users.ugent.beusers.ugent.be/~jdebeule/HonoldLandjev.pdf · “HonoldLandjev” — 2010/3/6 —...

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“HonoldLandjev” — 2010/3/6 — 16:04 — page 141 — #1 ii

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In: Current research topics in Galois geometryEditors: J. De Beule, L. Storme, pp. 141-167

ISBN 0000000000c© 2010 Nova Science Publishers, Inc.

Chapter 71

CODES OVER RINGS AND RING GEOMETRIES2

Thomas Honold∗†and Ivan Landjev‡§3

Abstract4

In this article, we bring together some recent results on special sets of points in5

coordinate projective geometries over finite chain rings. There is a clear coding theo-6

retic relevance of these results due to the strong connection between multisets of points7

in the chain ring geometries and so-called fat linear codes over finite chain rings. In8

Section 1, we introduce axiomatically projective and affine Hjelmslev spaces. An im-9

portant class of such spaces, obtained as coordinate geometries over finite chain rings,10

is given in Section 2. In Section 3, we define multisets of points in projective Hjelm-11

slev geometries and fat linear codes over finite chain rings. Furthermore, we state a12

result saying that these are essentially one and the same object. In Sections 4 and 5,13

we survey the known results on arcs and blocking sets in projective Hjelmslev planes.14

We include tables of the sizes of the largest known arcs in projective Hjelmslev planes15

over some small chain rings.16

Key Words: projective Hjelmslev geometry, projective Hjelmslev plane, finite chain ring,17

arcs, blocking sets, fat linear codes, Rédei type blocking sets, Witt vectors18

19

AMS Subject Classification: 51C05, 51E26, 51E21, 51E22, 94B05, 94B2720

1 Projective and affine Hjelmslev spaces21

?〈sec:hjelm〉?We start by introducing projective Hjelmslev spaces. The following axiomatic approach is22

due to Kreuzer [31–33, 35]. Let Π = (P ,L , I), I ⊆ P ×L , be an incidence structure. The23

∗Department of Information and Electronic Engineering, Zhejiang University, 38 Zheda Road, 310027Hangzhou, China. E-mail address: [email protected]

†Supported by the Open Project of Zhejiang Provincial Key Laboratory of Information Network Technologyand by the National Natural Science Foundation of China under Grant No. 60872063.

‡New Bulgarian University, 21 Montevideo str., 1618 Sofia, Bulgaria, and Institute of Mathematics andInformatics, BAS, 8 Acad. G. Bonchev str., 1113, Sofia, Bulgaria. E-mail address: [email protected]

§Supported by the Project Combined algorithmic and theoretical study of combinatorial structures betweenthe Research Foundation Flanders (Belgium) (FWO) and the Bulgarian Academy of Sciences, as well as by theStrategic Development Fund of the New Bulgarian University under Contract 357/14.05.2009.

mailto:[email protected]

mailto:[email protected]

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142 T. Honold and I. Landjev

sets P and L are referred to as sets of points and lines, respectively. A neighbour relation1

_ is defined on P and L by the following conditions:2

(N1) ∀x,y ∈ P : x _ y ⇐⇒∃S,T ∈ L ,S 6= T : {(x,S),(x,T ),(y,S),(y,T )} ⊆ I;3

(N2) ∀S,T ∈ L : S _ T ⇐⇒ for every point x with (x,S) ∈ I there is a point y with4

(y,T ) ∈ I and x _ y, and, conversely, for every y with (y,T ) ∈ I there is a point x with5

(x,S) ∈ I and y _ x.6

Given two points x,y with x 6 _ y we denote by x,y the unique line incident with both7

of them if such a line does exist. For a point x and a line S, we write x _ S if there exists a8

point y with (y,S) ∈ I, x _ y.9

Definition 1.1. An incidence structure Π = (P ,L , I) with neighbour relation _ is said to10

be a projective Hjelmslev space if it satisfies the following axioms:11

(H1) For any two points x,y ∈ P there exists a line S with (x,S) ∈ I, (y,S) ∈ I.12

(H2) Every line S ∈ L contains at least three points which are pairwise non-neighbours.13

(H3) Two lines S and T with S∩T 6= /0 are neighbours iff |S∩T | ≥ 2.14

(H4) For any x,y,z ∈ P , x _ y and y _ z imply x _ z.15

(H5) For any two lines S,T and any three points x,y,z with (x,S) ∈ I, (y,S) ∈ I, (x,T ) ∈ I,16

(z,T ) ∈ I, x 6_ y, x 6_ z, y _ z, we have S _ T .17

(H6) For a point x not incident with S ∈ L with x _ S, there always exist y,z ∈ P with18

y 6_ S, (z,S) ∈ I and (x,y,z) ∈ I.19

(H7) Let x∈ P , S∈L with x 6_ S and let y,z∈ S. For every (y′,x,y)∈ I and every (z′,x,z)∈20

I there exists a line T with (y′,T ) ∈ I, (z′,T ) ∈ I and S∩T 6= /0.21

The point set T ⊆ P is called a Hjelmslev subspace of Π if for every two distinct points22

x,y ∈ P , there exists a line L ∈ L(T ) = {L ∈ L | L ⊆ T } with (x,L) ∈ I, (y,L) ∈ I. We23

write x _ T if there exists a point y ∈ T with x _ y. Every Hjelmslev subspace T forms a24

projective Hjelmslev space (T ,L(T ), IT ) of its own, where IT = I∩(T ×L(T )). For every25

X ⊆ P we define the hull 〈X 〉 as the intersection of all Hjelmslev subspaces containing X .26

The set X ⊆ P is said to be independent if for any x ∈ X we have x 6_ 〈X \{x}〉.27

Definition 1.2. The point set B is a basis of Π if 〈B〉= P and B is independent.28

The dimension of a projective Hjelmslev space Π is defined as dimΠ = |B|−1. In what29

follows, we consider only finite-dimensional Hjelmslev spaces.30

2 Coordinate Hjelmslev geometries31

?〈sec:galois〉?An important class of projective Hjelmslev spaces can be obtained as coordinate geometries32

from modules over so-called finite chain rings. We review only the most basic properties of33

this class of finite rings, and refer the reader for a detailed treatment to [7, 39, 40, 42].34

An associative ring R with identity (1 6= 0) is called a left (right) chain ring if the lattice35

of left (resp., right) ideals of R forms a chain. In the finite case, |R| < ∞, this condition is36

left-right symmetric and equivalent to R being a local principal ideal ring. In what follows37

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Codes over rings and ring geometries 143

the Jacobson radical rad(R) of R (which we assume to be a finite chain ring from now1

on) will be denoted by N, so that R/N ∼= Fq is a finite field and N = Rθ = θR for any2

θ ∈ N \N2. Furthermore, there exists an integer m ≥ 1 (called the length or nilpotency3

index of R) such that Nm−1 6= {0}, Nm = {0}, and every left or right ideal of R belongs4

to the chain R > N > N2 > · · · > Nm−1 > {0} of two-sided ideals Ni = Rθi = θiR. The5

finite chain rings of length m = 1 are just the fields Fq and thus trivial from our point of6

view. For the smallest non-trivial case m = 2, a detailed description and classification of the7

corresponding rings will be given in Section 4.8

Let MR be a finite free (right) module over R of rank rkM ≥ 3. Denote by P and L9

the set of all free rank 1, respectively free rank 2, submodules of MR and by I ⊆ P ×L set-10

theoretical inclusion. The incidence structure (P ,L , I) satisfies (H1)–(H7) and, therefore, is11

a projective Hjelmslev space. If rkM = k, this incidence structure is referred to as the (right)12

(k−1)-dimensional projective Hjelmslev geometry over the chain ring R and is denoted by13

PHG(RkR). (Since MR ∼= Rk

R, this is no essential restriction.)14

Let R be a chain ring with |R| = qm, R/N ∼= Fq. We consider the projective Hjelmslev15

space Π = (P ,L , I) = PHG(RkR). Two points x = xR and y = yR are called i-neighbours,16

i = 0,1, . . . ,m, if |x∩ y| ≥ qi. This fact is denoted by x _ iy. Two lines S and T are i-17

neighbours if for every point x on S there exists a point y on T with x _ iy, and conversely,18

for every y on T there exists x on S with y _ ix. Every two points (lines) are 0-neighbours;19

1-neighbourhood is the same as the neighbour relation defined by (N1) and (N2).20

For every i ∈ {0,1, . . . ,m}, the relation _ i is an equivalence relation on P , as well21

as on L . The equivalence classes of this relation are denoted by [x](i), x ∈ P , respectively22

[S](i), S ∈ L . The set of all equivalence classes of _ i on points, resp. lines, is denoted23

by P (i), resp. L(i). We denote by π(i) the natural homomorphism π(i) : R → R/Rθi, where24

Rθ = radR. By π(i), we denote the mapping induced by π(i) on the Hjelmslev subspaces of25

Π.26

Below we state some facts on the combinatorics and the structure of the projective27

Hjelmslev geometries PHG(RkR) (cf. [2, 10, 12, 19, 29–31, 33, 44]).28

?〈fact:thm1〉?Fact 2.1. Let Π = (P ,L , I) = PHG(RkR), where R is a chain ring with |R|= qm, and R/N ∼=29

Fq. For every two integers s, t with 0 ≤ t ≤ s ≤ k, the number of all (s− 1)-dimensional30

Hjelmslev subspaces through a fixed (t−1)-dimensional subspace is equal to31

q(s−t)(k−s)(m−1)[

k− ts− t

]q,32

where33 [ks

]q=

(qk−1)(qk−1−1) · · ·(qk−s+1−1)(qs−1)(qs−1−1) · · ·(q−1)

.34

Moreover, the number of points that are i-th neighbours to a fixed point is q(k−1)(m−i) for all35

i = 1, . . . ,m.36

The next few results explain the structure of the geometries PHG(RkR) in some more37

detail.38

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Define a new incidence relation J(i) ⊆ P (i)×L(i) by1

([x](i), [S](i)) ∈ J(i) ⇔∃x′ ∈ [x](i),∃S′ ∈ [S](i) : (x′,S′) ∈ I.2

?〈fact:thm0〉?Fact 2.2. The incidence structure (P (i),L(i),J(i)) is isomorphic to the projective Hjelmslev3

geometry PHG((R/Rθi)kR/Rθi). In particular, (P (1),L(1),J(1)) is isomorphic to PG(k−1,q).4

Let ∆1,∆2 be two Hjelmslev subspaces with dim∆1 ≤ dim∆2. We write ∆1 _ i∆2 if5

π(i)(∆1) ⊆ π

(i)(∆2). Note that under this definition _ i is not symmetric. We consider6

again Π = (P ,L , I) = PHG(RkR), where R is a chain ring with |R|= qm, R/N ∼= Fq. Let us7

fix a Hjelmslev subspace Σ with dimΣ = u−1 and an integer i, 1 ≤ i ≤ m−1. Denote by8

Pi(Σ) the set of all points x with x _ iΣ. Now set9

P ={

∆∩ [x]m−i | x ∈ Pi(Σ),dim∆ = u−1, ∆ _ iΣ, ∆∩ [x]m−i 6= /0

}. (1)10

It can be proved that the sets ∆∩ [x]m−i are either disjoint or coincide for the various choices11

of ∆.12

Let S ∈ L . We say that the “point” x = ∆∩ [x]m−i ∈P is incident with the line S if13

∆∩ [x]m−i∩S 6= /0.14

This defines an incidence relation I′ ⊆ P×L . For two lines S and T we write S ∼ T if S15

and T are incident with the same points of P. Clearly ∼ is an equivalence relation on L .16

Denote by L a set of representatives from the different equivalence classes of lines under17

∼, which have nonempty intersection with at least one of the sets ∆∩ [x]m−i. Let J be the18

incidence relation induced by I′ on P×L. With the above notation, we have the following19

result.20

〈fact:thm3〉Fact 2.3. The incidence structure (P,L,J) can be embedded isomorphically into21

PHG((R/Rθm−i)kR/Rθm−i). The missing part consists of the points of a (k − u − 1)-22

dimensional Hjelmslev subspace.23

The (k− 1)-dimensional affine Hjelmslev geometry AHG(Rk−1R ) is defined as the in-24

cidence structure obtained from PHG(RkR) by deleting a neighbour class of hyperplanes.25

Equivalently, it can be defined as the incidence structure having as points all (k−1) tuples26

(α1, . . . ,αk−1), αi ∈ R, and as lines all cosets of free rank 1 submodules of Rk−1R . If in the27

discussion preceding Fact 2.3, we take Σ to be a point, say x, then P = [x]i and we get the28

following result.29

?〈fact:thm2〉?Fact 2.4. ([x]i,L,J)∼= AHG((R/Rθ

m−i)k−1R/Rθm−i

).30

In particular,31

([x]m−1,L)∼= AG(k−1,q).32

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3 Multisets of points in projective Hjelmslev geometries and lin-1

ear codes over finite chain rings2

?〈sec:codmul〉?3.1 Multisets of points in PHG(RkR)3

?〈ssec:multisets〉?Let Π = PHG(RkR) = (P ,L , I) be a finite-dimensional projective Hjelmslev geometry over4

the chain ring R.5

Definition 3.1. A multiset in Π is a mapping K : P → N0.6

The mapping K is extended to the subsets of P by7

K(Q ) = ∑x∈Q

K(x) for Q ⊆ P . (2)8

The integer K(x) is called the multiplicity of the point x. The integer K(P ) = ∑x∈P K(x)9

is called the cardinality or length of the multiset K and is denoted by |K|. The support10

suppK of K is defined by suppK = {x ∈ P |K(x) > 0}. For a multiset K in Π we define its11

hull 〈K〉 ≤ RkR by12

〈K〉= ∑xR∈suppK

xR. (3)13

Clearly, 〈K〉 can be considered as the set of all points x = xR with x ≤ 〈K〉.14

Given a set of points Q ⊆ P , we define the characteristic multiset χQ by15

χQ (x) ={

1 if x ∈ Q0 otherwise.

16

All multisets K satisfying K(x) ∈ {0,1} for every x ∈ P arise in this manner from their17

supports. Such multisets are said to be projective and may be tacitly identified with their18

supports.19

The multiset K induces in a natural way multisets K(i) in π(i)(Π) by20

K(i) : P (i) → N0 : [x]i 7→ K([x]i)21

for i = 0,1, . . . ,m. Note that π(i)(〈K〉) = 〈K(i)〉.22

〈l:rank〉Definition 3.2. Denote by κi the rank of the R-module 〈K(i)〉.23

In geometric language, κi − 1 is the dimension of the smallest Hjelmslev subspace of24

π(i)(Π) containing all points of suppK(i).25

?〈dfn:mset-equiv〉?Definition 3.3. Two multisets K in Π and K′ in Π′ are said to be equivalent if there exists26

a bijective R-semilinear mapping ψ : 〈K〉R → 〈K′〉R such that K(x) = K′(ψ(x))

for every27

point x ∈ 〈K〉.28

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3.2 Linear codes over finite chain rings1

?〈ssec:codes〉?Let R be a chain ring with |R| = qm, R/N ∼= Fq, let θ be a generator of N, and consider2

the set Rn of all n-tuples over R. The set Rn has the structure of an (R-R)-bimodule with3

respect to component-wise addition and left/right multiplication by elements from R. We4

say that θi is the period of the vector x ∈ Rn if i is the smallest non-negative integer with5

θix = 0 (equivalently, with x ∈ Rnθm−i). We denote this by θm−i ‖ x. The set of vectors in6

Rn of period θm is denoted by (Rn)∗. Since Rθi = θiR for all i ≥ 0, the concept of period is7

left-right symmetric even for non-commutative chain rings.8

?〈dfn:code〉?Definition 3.4. A code C of length n over R is a non-empty subset of Rn. The vectors of C9

are called codewords. The code C is left (resp., right) linear if it is an R-submodule of RRn10

(resp., of RnR). A linear code is one which is either left or right linear.11

Definitions and results in the sequel will be stated for left linear codes, most of them12

having obvious right counterparts.13

A partition λ ` n of an integer n is a sequence of non-negative integers λ0 ≥ λ1 ≥14

λ2 ≥ . . . with ∑i≥0 λi = n. The trailing zeros of this sequence will be suppressed. The15

following theorem generalizes the structure theorem for finite abelian p-groups (see e.g. [38,16

Ch. 15,§ 2]):17

?〈thm:cyclic〉?Theorem 3.5 ( [21]). Every linear code C over a chain ring R is a direct sum of cyclic18

R-modules. The partition λ = (λ1, . . . ,λk) ` logq|C | satisfying19

RC ∼= R/Nλ1 ⊕·· ·⊕R/Nλk (4)20

is uniquely determined by RC . Moreover, the partition µ = λ′ ` logq|C | conjugate to λ has21

components µi = dimR/N(θi−1C/θiC ).22

?〈dfn:shape〉?Definition 3.6. The shape of a linear code C over R is the partition

λ = (λ1, . . . ,λk) ` logq|C |,

which satisfies RC ∼= R/Nλ1 ⊕ ·· · ⊕R/Nλk . The partition λ′ conjugate to λ is called the23

conjugate shape of C . The integer k = λ′1 = dimR/N(C/θC ) is called the rank of C and is24

denoted by rkC . A subset {x1, . . . ,xk} ⊆ C \{0} is called a basis of C if RC = Rx1⊕·· ·⊕25

Rxk.26

?〈dfn:gmatrix〉?Definition 3.7. Let C ≤ RRn be a linear code of rank rkC = k. A generator matrix of C is27

a k×n-matrix having as its rows a basis of C , so that, in particular, C = {xG;x ∈ Rk}.28

For two vectors u = (u1, . . . ,un) ∈ Rn and v = (v1, . . . ,vn) ∈ Rn we define their inner29

product u ·v by30

u ·v := u1v1 + · · ·+unvn. (5)31

Given a code C ⊆ Rn, we define32

C⊥ = {y ∈ Rn | x ·y = 0 for every x ∈ C},33

⊥C = {y ∈ Rn | y ·x = 0 for every x ∈ C}.34

The linear code C⊥ ≤ RnR (resp., ⊥C ≤ RRn) is called the right (resp., left) orthogonal code35

of C .36

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〈thm:perp〉Theorem 3.8 ( [21]). Let C ≤ RRn be a linear code of shape λ = (λ1, . . . ,λn) and rank1

λ′1 = k.2

〈l:perp-shape〉1. The orthogonal code C⊥ has shape (m− λn,m− λn−1, . . . ,m− λ1) and conjugate3

shape (n−λ′m,n−λ′m−1, . . . ,n−λ′1). In particular, C is free as an R-module if and4

only if C⊥ is free if and only if rk(C⊥) = n− k;5

〈l:perp-perp〉2. ⊥(C⊥) = C ;6

?〈l:perp-lattice〉?3. if in addition C ′ ≤ RRn then (C ∩C ′)⊥ = C⊥+C ′⊥, and (C +C ′)⊥ = C⊥∩C ′⊥.7

?〈cor:row-column〉?Corollary 3.9. Let G ∈ Mm,n(R) be any matrix. The linear codes C ≤ RRn and D ≤ RmR8

generated by the rows and columns of G, respectively, have the same shape.9

?〈dfn:kmatrix〉?Definition 3.10. A parity check matrix of a linear code C ≤ RRn is an (n−λ′m)×n-matrix10

whose rows form a basis of the orthogonal code C⊥11

Note that if H is a parity-check matrix of C , then by Part 2 of Theorem 3.8 we have12

x ∈ C if and only if x ·HT = 0. The number of (and periods of the) columns of H are13

determined by Part 1 of Theorem 3.8.14

For x = (x1, . . . ,xn) ∈ Rn we set15

ai(x) = |{ j | 1 ≤ j ≤ n and θi ‖ x j}|.16

?〈dfn:type〉?Definition 3.11. The sequence(a0(x), . . . ,am(x)

)is called the type of the word x ∈ Rn.17

?〈dfn:isom〉?Definition 3.12. An automorphism of the code Rn is a bijective mapping φ : Rn → Rn which18

satisfies ai(x−y) = ai(φ(x)−φ(y)

)for all x,y ∈ Rn and all i ∈ {0,1, . . . ,m}.19

?〈dfn:cisom〉?Definition 3.13. Two codes C1,C2 ⊆ Rn are said to be isomorphic (resp., semilinearly iso-20

morphic) if there exists a code automorphism (resp., semilinear code automorphism) φ of21

Rn with φ(C1) = C2.22

3.3 Equivalence of multisets of points and linear codes23

?〈ssec:equiv〉?Definition 3.14. A linear code C ≤ RRn is said to be fat if for every i ∈ {1, . . . ,n} there24

exists a codeword c = (c1,c2, . . . ,cn) ∈ C with ci ∈ R× (i.e. ci is a unit in R).25

Let C ≤ RRn be a fat linear code. Let S = (c1, . . . ,ck) be a sequence of (not necessarily26

independent) generators for RC and let G∈Mk,n(R) be the k×n-matrix with rows c1, . . . ,ck.27

Denote the columns of G by g1, . . . ,gn. Since C is fat and c1, . . . ,ck generate C , the vectors28

g j have period θm and thus define points g jR in the projective (right) Hjelmslev geometry29

(P ,L ,I ) = PHG(RkR). We define the multiset KS induced by the generating sequence S of30

C as31

KS : P → N0 : x 7→ |{ j | x = g jR}|. (6)32

We say that the multiset KS and the code C = Rc1 + · · ·+ Rck are associated. By the33

definition of KS, we have |KS| = n. Furthermore, the modules 〈KS〉 and RC have the same34

shape and, in particular, the same cardinality; see [21].35

The following theorem is a generalization of a similar result by Dodunekov and Simonis36

[11] about linear codes over finite fields.37

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?〈thm:codmul〉?Theorem 3.15. For every multiset K of length n in PHG(RkR) there exists a fat linear code1

C ≤ RRn and a generating sequence S = (c1, · · · ,ck) of C which induces K. Two multisets K12

in PHG(Rk1R ) and K2 in PHG(Rk2

R ) associated with fat (left) linear codes C1 and C2 over R,3

respectively, are equivalent if and only if the codes C1 and C2 are semilinearly isomorphic.4

?〈dfn:ktype〉?Definition 3.16. Let K : P → N0 be a multiset in Π = PHG(RkR). A hyperplane ∆ in Π is5

said to have the K-type (a0,a1, . . . ,am), where6

ai = ∑x _ i∆,x 66_ i+1∆

K(x) for0 ≤ i ≤ m.7

By duality (cf. Theorem 3.8), every hyperplane ∆ in PHG(RkR) can be considered as a

set of points, whose homogeneous coordinates (x1, . . . ,xk) satisfy a linear equation

r1x1 + r2x2 + . . .+ rkxk = 0,

where at least one of the ri’s is a unit in R. Let C be a fat linear code associated with K,8

and let GS be a k× n-matrix whose sequence S of rows generates C and satisfies KS =9

K. All codewords of C which belong to the cyclic submodule R(r1, . . . ,rk)GS ≤ RC are10

called codewords associated with the hyperplane ∆ (relative to the choice of the generating11

sequence S). There is a connection between the K-type of a hyperplane in Π and the number12

of codewords of a given type in C associated with that hyperplane.13

?〈thm:nowords〉?Theorem 3.17. Let K be a multiset in PHG(RkR) and let C be a fat linear code over R14

associated with K. For each hyperplane ∆ of K-type (0, . . . ,0,a j,a j+1, . . . ,am), with a j 6= 0,15

0 ≤ j ≤ m, there exist exactly qm−s−qm−s−1 codewords in C of type16

(0, . . . ,0︸︷︷︸s

,a j, . . . ,am+ j−s−1,m

∑i=m+ j−s

ai) ( j ≤ s ≤ m−1) (7)17

which are associated with ∆.18

For a multiset K in PHG(RkR), the numbers κi = rk〈K(i)〉 (Definition 3.2) determine the19

shape of every fat linear code C ≤ RRn associated with K.20

〈thm:card〉Theorem 3.18. Let K be a multiset in PHG(RkR) associated to the fat linear code C . Then

C has conjugate shape λ′ = (κm,κm−1, . . . ,κ1), and, in particular,

|C |= qκ1+κ2+···+κm .

3.4 Some classes of codes defined geometrically21

?〈ssec:geocodes〉?Consider the Hjelmslev geometry Π = (P ,L , I) = PHG(RkR). The linear code C associated22

with the multiset K defined by K(x) = 1 for all x ∈ P , is called the k-dimensional simplex23

code over R and is denoted by Sim(k,R). The code Sim(k,R) has length q(k−1)(m−1)[k

1

]q,24

and by Theorem 3.18 it has shape mk (i.e. its shape consists of k parts equal to m), in25

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particular |Sim(k,R)|= qkm. All hyperplanes ∆ in Π have the same K-type (a0,a1, . . . ,am),1

where2

a0 = q(k−1)(m−1)

([k1

]q−[

k−11

]q

)= q(k−1)m,

a j = q(k−2)(m−1)[

k−11

]q

(qm− j−qm− j−1) , j = 1, . . . ,m−1,

am = q(k−2)(m−1)[

k−11

]q.

The dual code Sim(k,R)⊥ is called the k-th order Hamming code over R and is de-3

noted by Ham(k,R). It is free of rank q(k−1)(m−1)[k

1

]q − k, in particular |Ham(k,R)| =4

qmq(k−1)(m−1)[k1]q−mk. A parity check matrix and a generator matrix for Ham(k,R) may be5

obtained similarly to the special case of R being a field.6

4 Arcs in projective Hjelmslev planes7

〈sec:arcs〉4.1 The maximal arc problem8

?〈ssec:maxarcpbm〉?〈dfn:arcs〉Definition 4.1. A multiset K in (P ,L , I) is called a (k,n)-arc if9

(i) K(P ) = k.10

(ii) K(L)≤ n for every line L ∈ L .11

According to this definition, a (k,n)-arc is also a (k,n′)-arc for every integer n′ ≥ n. For12

this reason we shall usually assume that n is chosen to be minimal, i.e. there exists at least13

one line L0 ∈ L with K(L0) = n (but there are exceptions). Moreover, sometimes we say14

“n-arc” in place of “(k,n)-arc” without referring to the cardinality of K.15

Of course, Definition 4.1 also makes sense for other incidence structures. In the clas-16

sical cases of PG(2,q) or AG(2,q) (which can be considered as special cases of projective17

Hjelmslev planes) a lot of research has been done on arcs and many results are known. For18

an overview, see, for example [9]. Some of these results will be used in the sequel.19

The arcs considered in this section will be projective and can be identified with sets of20

points, as described earlier.21

Furthermore, for the rest of this survey, we will confine ourselves to the case of finite22

chain rings R of length 2, i.e. the case |R| = q2, R/N ∼= Fq. The classification of all those23

rings is known and summarized in the following result.24

?〈fact:classify〉?Fact 4.2 ( [43, Th. 4] or [8, Th. 6]). Suppose R is a finite chain ring with |R|= q2, R/N ∼= Fq,25

where q = pr. Then26

(i) either R has characteristic p2 and is isomorphic to the Galois ring GR(q2, p2) of order27

q2 and characteristic p2, defined as GR(q2, p2) = Zp2 [X ]/(h) where h ∈ Zp2 [X ] is a28

(monic) polynomial of degree r which is irreducible modulo p, or29

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(ii) R has characteristic p and for some σ ∈ Aut(Fq) is isomorphic to the ring1

Fq[X ;σ]/(X2) of σ-dual numbers over Fq, defined as the set of all a0 + a1X ∈ Fq[X ]2

with operations (a0 + a1X) + (b0 + b1X) := a0 + b0 + (a1 + b1)X, (a0 + a1X)(b0 +3

b1X) := a0b0 +(a0b1 +a1σ(b0)

)X.4

Moreover, the r +1 different rings listed in (i), (ii) are pairwise non-isomorphic.5

In the sequel, we will also refer to these rings as Gq := GR(q2, p2), Sq := Fq[X ]/(X2),6

and T(i)q := Fq[X ;σi]/(X2) for 1 ≤ i ≤ r−1, where σ denotes the Frobenius automorphism7

of Fq. Furthermore we will use the abbreviations Tq = T(1)q and T◦

q = T(r−1)q .18

Denote by mn(R3R) the maximal value of k for which there exists a (k,n)-arc in9

PHG(R3R). The problem of determining the exact values of mn(R3

R) for various values of10

n and for various rings R is central and has a clear coding theoretic relevance.11

4.2 A general upper bound on the size of an arc12

?〈ssec:upperbound〉?The following theorems provide upper bounds on the size of a (k,n)-arc in PHG(R3R) [37].13

?〈thm:bound〉?Theorem 4.3. Let K be a (k,n)-arc in PHG(R3R) where |R| = q2, R/N ∼= Fq. Suppose14

there exists a neighbour class of points [x] with K([x]) = u and let ui, i = 0,1, . . . ,q, be the15

maximum number of points on a line from the i-th parallel class in the affine plane defined16

on [x]. Then17

k ≤ q(q+1)n−qq

∑i=0

ui +u.18

Proof. Let {Li | i = 0,1, . . . ,q} be a set of q + 1 lines no two of which are neighbours andsuch that K([x]∩Li) = ui. For every i ∈ {0, . . . ,q}, denote by L( j)

i , j = 1, . . . ,q, the q linesin PHG(R3

R) that coincide with Li on [x]. The sum of the multiplicities of the points fromL( j)

i not in [x] does not exceed n−ui, which gives the estimate

k = K([x])+q

∑i=0

q

∑j=1

K(L( j)i \ ([x]∩Li))

≤ u+q

∑i=0

q

∑j=1

(n−ui)

= u+q

∑i=0

q(n−ui)

= u+q(q+1)n−qq

∑i=0

ui.

19

1The latter reflects the fact that T(r−1)q is isomorphic to the opposite ring of T(1)

q . Note that the smallest case

where a symbol T(i)q cannot be avoided is q2 = 256.

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Typically, the numbers ui are unknown. We can use some simple estimates to get a1

more convenient form for the above upper bound. From the obvious inequality ui ≥ du/qe,2

we get3

k ≤ q(q+1)(n−du/qe)+u. (8)4

Fix a point y ∈ [x] and let S0, . . . ,Sq be lines through y, no two of which are neighbours.Without loss of generality, we assume that Li _ Si for i = 0, . . . ,q. Set si = K([x]∩ Si)−K(y). Clearly, K(y)+ si ≤ ui. Then

k ≤ q(q+1)n−qq

∑i=0

ui +u

≤ q(q+1)n−qq

∑i=0

(K(y)+ si)+u

= q(q+1)n−q(q+1)K(y)−q(u−K(y))+u

= q2(n−K(y))+q(n−u)+u.

Since we may certainly assume that K(y)≥ 1, the last inequality simplifies to5

k ≤ q2(n−1)+q(n−u)+u.6

〈thm:genbound〉Theorem 4.4.

mn(R3)≤ max1≤u≤min{µn(q),q2}

min{u(q2 +q+1),

q2(n−1)+q(n−u)+u,q(q+1)(n−du/qe)+u},

where µn(q) denotes the maximal size of a (k,n)-arc in AG(2,q).7

For small values of n, we can derive somewhat better bounds.8

?〈thm:oval_bound〉?Theorem 4.5. m2(R3)≤

{q2 +q+1 for q even,q2 for q odd.

(9)9

In case of equality, we have10

(i) for q even, K([x]) = 1 for every [x] ∈ P (1);11

(ii) for q odd, K([x]) ≤ 1 for every [x] ∈ P (1). Moreover, the neighbour classes with12

K([x]) = 0 form a line in the factor plane (P (1),L(1),J (1))∼= PG(2,q).13

?〈thm:q=9〉?Theorem 4.6. m3(R3R)≤ 2q2−q+3, for every q ≥ 5.14

Note that in the cases q = 2,3, the exact value of m3(R3R) is known. It is 10 for the rings15

of cardinality 4, 19 for R = Z9, and 18 for R = F3[X ]/(X2). For q = 4, we have the bounds16

29 ≤ m3(R3)≤ 30 for all three rings G4, S4, T4.17

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4.3 Constructions for arcs1

?〈ssec:constructions〉?In this section, we present general constructions for arcs in projective Hjelmslev planes.2

Throughout this section, R will be a chain ring with |R|= q2, R/N ∼= Fq, and Π = (P ,L , I)3

will be the projective Hjelmslev plane PHG(R3R).4

〈ex:ex1〉Example 4.7. For values of n close to q2 + q, the exact value of mn(R3R) can be easily5

computed. For every chain ring R, with |R|= q2, R/N ∼= Fq, and every integer s = 0,1, . . . ,q6

mq2+s(R3R) = q4 +q2s+qs.7

Denote by F the point set obtained in the following way. Fix a line L. Take in F the8

points of the line L plus q− s− 1 additional line segments parallel to L∩ [xi] in each of9

the neighbour classes [xi] incident with [L] in the factor geometry. The multiset χP −χF is10

easily checked to be the desired arc. The upper bound is obtained from Theorem 4.4.11

Example 4.8. Now we describe a general construction for (q4− q2− 2q + 1,q2− 1)-arcs12

in PHG(R3R) that does not depend on the underlying ring. Remarkably, this construction13

is better than the “triangle construction” which yields a (q4−2q2 +1,q2−1)-arc χP −χF14

as the complement of a “triangle” F (F consists of a neighbour class of lines and two15

additional lines that are not neighbours).16

Fix a point class [x0] and a line class [L0] incident with [x0] in the factor plane. Set17

[L0] = {[xi] | i = 0, . . . ,q}.18

Furthermore, denote by [Li], 1≤ i≤ q, the other line classes through [x0] in the factor plane.19

Consider the set K containing the following points:20

1) The complement of a (2q−1)-blocking set in the affine plane induced on [x0] (which is21

isomorphic to AG(2,q)). Thus K contains (q−1)2 points from [x0].22

2) The line segments from the point classes [x1], . . . , [xq] together with q additional lines23

(containing the segments in [xi], i = 1, . . . ,q) form a structure isomorphic to AG(2,q).24

In every class [xi], choose q−2 line segments (having the direction of [L0]) such that the25

resulting q(q−2) line segments form the complement of a blocking set in AG(2,q).26

3) From each of the remaining point classes [y], select the following points. If [y] ∈ [Li],27

take the q2−q points from q−1 parallel line segments having the direction of the line28

[yxi].29

The total number of points is (q− 1)2 + q · q(q− 2) + q2(q2 − q) = q4 − q2 − 2q + 1.30

A line in [L0] meets [x0]∩K in at most q− 1 points and at most q− 1 of the sets [xi]∩K,31

i = 1, . . . ,q, in q points, i.e., it contains at most q−1+(q−1)q = q2−1 points from K.32

A line in the class [yx0], y 6_ L0, meets [x0]∩K in at most q−1 points and each of the33

other q sets [z]∩K in exactly q−1 points. Hence, such a line contains at most q−1+q(q−34

1) = q2−1 points from K.35

Finally, a line in the class [yxi], y 6_ L0, i 6= 0, meets one set [z]∩K in at most q points,36

q−1 such sets in q−1 points and one set (the set [xi]∩K) in q−2 points. Therefore, such37

a line contains at most q+(q−1)2 +q−2 = q2−1 points.38

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Thus the arc defined above has the desired parameters.1

This construction can be further improved if we take the blocking set on [L0] to consist2

of two lines that meet on [x0]. Furthermore, we replace the q−1 points from [x0] that form3

a line segment in a direction different from that of L0 by q− 2 collinear points in [x1] that4

again have a direction different from that of L0 and are not already part of the blocking5

set on [L0]. It is an easy check that the size of the arc is increased by 1 and we get a6

((q3 +q2−2)(q−1),q+1)-arc.7

For q2 = 9,16,25, this construction gives: m8(R3R)≥ 68, for q2 = 9,m8(R3

R)≥ 234, for8

q2 = 16, and m8(R3R)≥ 592, for q2 = 25.9

The exact formula for mn(R3R) in the range q2 ≤ n ≤ q2 + q presented in Example 4.710

may also be written as mn(R3R) = q4 + q3 + q2 − (q2 + q− n)(q2 + q). From this point of11

view it says that the complementary((q2 + q− n)(q2 + q),q2 + q− n

)-blocking set has12

the same cardinality as the (generally non-projective) sum of q2 + q− n lines. It seems13

reasonable to conjecture that the lower bound mn(R3R)≥ q4 +q3 +q2− (q2 +q−n)(q2 +q)14

holds for all n. (For small values of n, this lower bound is even rather weak.) The following15

theorem extends the range of integers n, for which the lower bound is known to hold, to16

q2−bq/2c ≤ n ≤ q2 +q.17

?〈thm:triangle〉?Theorem 4.9. For every chain ring R with |R| = q2, R/ radR ∼= Fq, and every integer s =18

1,2, . . . ,bq/2c, the following inequality holds:19

mq2−s(R3R)≥ q4−q2s−qs. (10)20

Proof. We will prove the existence of a(t(q2 +q), t

)-blocking set in PHG(R3

R) for q+1 ≤21

t ≤ b3q/2c except in the case (q, t) = (3,4), which is covered by the subsequent Exam-22

ple 4.10.23

Choose point classes [x0], [x1], [x2] and line classes [L0], [L1], [L2] which form a triangle24

in the factor plane PG(2,q), indexed in such a way that [xi] is incident with [Li−1] and [Li]25

(indices taken modulo 3). There exist (unique) integers t1, t2, t3 satisfying t = t1 + t2 + t326

and 1 ≤ t1 ≤ t2 ≤ t3 ≤ t1 + 1. In each point class [x] incident with [Li] but different from27

the vertices [xi] and [xi+1], choose ti parallel line segments in the direction of [Li]. In each28

class [xi] choose ti−1 + ti parallel line segments in the direction of [Li]. This is possible,29

since ti−1 + ti ≤ t − t1 = t −bt/3c = d2t/3e ≤ q. It is clear that the resulting point set in30

PHG(R3R) blocks every line outside [L1]∪ [L2]∪ [L3] exactly t times. 2 Every line L ∈ [Li] is31

blocked ti +ti+1 times by the line segments in [xi+1]. Since t3 = dt/3e ≤ dq/2e< q, we have32

q+ ti + ti+1 > t. In order to have K(L)≥ t, it is therefore enough to ensure that L is blocked33

at least once by the line segments chosen in [Li]\ [xi+1]. The q2 line segments in [Li]\ [xi+1]34

(as points) together with the q2 lines in [Li] and the q point classes [y1] = [xi], [y2], . . . , [yq] (as35

lines) form an incidence structure isomorphic to AG(2,q). Our task is to arrange the ti−1 +ti36

line segments in [y1] and the ti line segments in each class [y j], 2 ≤ j ≤ q, in such a way37

that they form a blocking set in AG(2,q).3 Since ti ≥ 1, we may assume that q of these line38

2The construction so far can also be seen as taking the sum of t = t1 + t2 + t3 lines in PHG(R3R), where

ti lines are chosen from [Li] in such a way that they have a line segment in [xi+1] in common. To make theresulting multiset projective, the ti-fold line segment in [xi+1] is replaced by ti line segments in [xi+1] havingdirection [Li+1] and not already chosen during the first step.

3Note that the special lines [y1], [y2], . . . , [yq] are blocked by construction, since ti−1 + ti > ti ≥ 1.

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segments, one from each class [y j], are collinear. The remaining ti−1 +ti−1+(q−1)(ti−1)1

line segments can be used to block the q−1 lines parallel to this line (and thus construct the2

required blocking set), provided there are at least q− 1 of them. If ti > 1, we are done. If3

ti = 1, then either (q, t) = (3,4) or (q, t) = (4,5). The first case has been already excluded.4

In the second case, we have t1 = 1, t2 = t3 = 2. We change the direction of the 3 line5

segments in [x1] from [L1] to [L0]. Then each line in [L1] is blocked 6 times, while for [L2],6

[L3] we have t2 > 1, t3 > 1 and thus are done.7

〈ex:q=3t=4〉Example 4.10. The following construction produces a (48,4)-blocking set in the projective8

Hjelmslev planes PHG(Z39) and PHG(S3

3), with S33 = F3[X ]/(X2). The factor plane PG(2,3)9

contains an oval (quadrangle) which has 4 tangents and 6 external points (intersection points10

of the tangents). Each external point is on exactly two tangents. In each point class [x]11

external to the oval, place a double line segment in one of the two tangent directions. Choose12

the directions in such a way that no tangent is chosen more than twice. In each point class13

[x] on the oval, place a single line segment in the tangent direction. For those tangent14

directions [T ] which were chosen twice in the above process, arrange the 5 line segments in15

[T ] with direction [T ] in such a way that they block every line in [T ]. As is easily verified,16

the resulting point set forms a (48,4) blocking set. The complementary (69,8)-arc was17

originally discovered during a computer search [28]. The computational data suggested the18

preceding construction.19

Example 4.11. The general cascade construction.20

The following general cascade construction has been proposed in [22]. Let K0 be a21

(k0,n0)-arc in PG(2,q). Let suppK = {x1, . . . ,xk0} and let {X1, . . . ,Xk0} be a set of k0 lines22

in PG(2,q) such that xi ∈ Xi. Then for each pair of integers α,s ∈ {1, . . . ,q}, there exists an23

arc in PHG(R3R) with parameters (αsk0,max0≤i≤k0 νi), where ν0 = αn0 and νi = s+α|Xi∩24

suppK0|−α, for i = 1, . . . ,k0.25

Below a special instance of this construction is described. Take q2 = 25, s = 5, and K026

to be an (11,3)-arc in PG(2,5). There exist two such arcs and for both of them a1 +a2 = 11.27

Select the lines Xi to be the 1- and 2-lines of K0. It is easily checked that max0≤i≤k0 νi =28

max{5+α,3α}. For α = 2, we get a (110,7)-arc, while for α = 3, we get a (165,9)-arc.29

Example 4.12. Take K0 to be the trivial (q2 + q + 1,q + 1)-arc in PG(2,q) consisting of30

all the points of the plane. Index the point and line classes in PHG(R3R) in such a way31

that [xi] is incident with [Li] in the factor geometry, i = 1, . . . ,q2 + q + 1. Select a line32

segment in each neighbour class consisting of q points that are collinear with a line from33

[Li]. Denote this set of points by F . The arc χF has parameters (q(q2 + q + 1),2q). More34

importantly, it gives rise to a strongly regular graph in the following way (as described35

in [6]). Let C ≤ RRn, n = q(q2 + q + 1), be a linear code associated with F . Since every36

line of PHG(R3R) is incident with either q or 2q points of F , there are only two F -types37

of lines, (a0,a1,a2) = (q3,q2,q) and (q3,q2 − q,2q), which in turn yield three types of38

non-zero codewords in C , namely(a0(x),a1(x),a2(x)

)= (q3,q2,q), (q3,q2 − q,2q) and39

(0,q3,q2 + q) with corresponding frequencies q5 − q2, (q− 1)(q5 − q2) and q3 − 1. Now40

take G = (V,E) as the Cayley graph of (C ,+) with respect to the set C1 ⊂ C of codewords41

of type (q3,q2−q,2q), i.e. G has vertex set V = C and edge set E ={(x,y) | x−y ∈ C1

}.442

4Since C1 =−C1, this actually defines an undirected graph.

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As shown in [6], the graph G is strongly regular with parameters1

v = q6, k = q5−q2, λ = q4 +q3−3q2, µ = q4−q2.2

Moreover, C can be mapped (cf. [20]) onto a (possibly non-linear) two-weight code over3

Fq.4

In the next section, we give an algebraic construction for (k,2)-arcs.5

4.4 (k,2)-Arcs6

〈ssec:n=2〉For (k,2)-arcs we have the bound (9). In some cases this bound is achieved. There exists7

a (7,2)-arc in the plane over G2 = Z4, but there is no such arc in the plane over S2 =8

F2[x]/(x2). There exist (9,2)-arcs in the projective Hjelmslev planes over both chain rings9

with 9 elements. For larger chain rings, it is possible to get large (k,2)-arcs with more than10

one point in some of the neighbour classes.11

Remarkably, (q2 + q + 1,2)-arcs exist in the planes over the Galois rings G2r =12

GR(4r,4) for all r. Below, we explain the construction in a more general setting [14,23,24].13

Let q = pr > 1 be a prime power and Gq = GR(q2, p2) be the Galois ring of cardinality14

q2 and characteristic p2. For any k ∈ N, the ring Gqk is the unique Galois extension of Gq15

of degree k and conversely, Gqk contains a unique subring isomorphic to Gq. It is known16

that Gqk is free of rank k as a module over Gq. Hence, Gqk can be viewed as the underlying17

module of the (k−1)-dimensional projective Hjelmslev geometry over Gq. We denote this18

geometry by PHG(Gqk/Gq).19

The group G×q of units of Gq contains a unique cyclic subgroup Tq of order q−1, called20

the group of Teichmüller units. This applies to both Gq and its extension ring Gqk , and we21

have Tqk = 〈η〉, Tq = 〈η(qk−1)/(q−1)〉 for any element η ∈G×qk of order qk−1.22

?〈dfn:teich〉?Definition 4.13. The set {Gη j | 0 ≤ j < (qk −1)/(q−1)} is called the Teichmüller set of23

PHG(Gqk/Gq) and is denoted by Tq,k.24

Since {η j | 0 ≤ j < (qk − 1)/(q− 1)} is a set of coset representatives for Tq in Tqk ,25

the Teichmüller set Tq,k contains exactly one point from each neighbour class. In case of26

G2 = Z4, k odd, the linear code over Z4 associated with T2,k (via the columns of a generator27

matrix) is isomorphic to the shortened quaternary Kerdock code; cf. [13, 41].28

Recall that a set of points is called a cap if no three points of this set are collinear.29

〈thm:teichcap〉Theorem 4.14. Let Gq = GR(q2, p2) be a Galois ring of characteristic p2 and let k ≥ 3 be30

an integer.31

- If every prime divisor of k is larger than p, then the Teichmüller set Tq,k is a cap in32

PHG(Gqk/Gq).33

- If k is even, Tq,k is never a cap.34

In particular, the Teichmüller set T2r,3 forms a (22r +2r +1,2)-arc in the projective Hjelm-35

slev plane PHG(G23r/G2r)∼= PHG(G32r) over the Galois ring G2r .36

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For projective Hjelmslev planes over chain rings R containing a subring isomorphic to1

the residue field of R, the following result holds [23].2

〈thm:ovalimp1〉Theorem 4.15. Let R be a chain ring with |R| = 22r, R/N ∼= F2r , charR = 2. Then there3

exists no (22r +2r +1,2)-arc in the projective Hjelmslev plane PHG(R3R).4

At present, it is not known whether (22r +2r,2)-arcs do exist over chain rings of nilpo-5

tency index 2 and characteristic 2, except for the two smallest cases. The answer is positive6

for q = 2, but negative for q = 4; see [26].7

For odd characteristics, the following theorem has been recently proved [16].8

〈thm:ovalimp2〉Theorem 4.16. Let R = Fq[X ;σ]/(X2) be a chain ring of length 2 and prime characteristic.9

There exists a (q2,2)-arc in the projective Hjelmslev plane PHG(R3R).10

A (21,2)-arc in the plane over Z25 has been constructed recently [27]. Below, we give11

a (21,2)-arc in PHG(Z325) taken from the online tables [1]. The points are represented by12

the columns of a 3×21-matrix over Z25.13 0 1 5 1 1 15 1 1 10 1 1 1 1 1 1 0 1 1 1 1 10 5 1 7 15 1 0 3 1 11 18 24 2 13 22 1 1 20 4 12 141 0 6 17 24 4 1 3 18 7 15 7 8 22 11 15 11 23 1 24 3

14

4.5 Dual constructions15

〈ssec:dual〉Let Π = (P ,L , I) = PHG(R3

R) be a coordinate projective Hjelmslev plane over a finite16

chain ring R. Using duality properties of the inner product R3×R3 → R: (x,y) 7→ x · y =17

x1y1 + x2y2 + x3y3, one can show that the dual plane Π∗ = (L ,P , I∗) is isomorphic to the18

left coordinate plane PHG(RR3) or, what is the same, to the projective Hjelmslev plane19

PHG(S3S) over the opposite ring S = R◦. This duality can be exploited in some cases for20

new constructions of arcs with good parameters.21

〈ex:dualarcs〉Example 4.17. There exist maximal((q4 − q)/2,q2/2

)-arcs in the projective Hjelmslev22

planes over the Galois rings Gq, q = 2r. These arcs are obtained by taking K as the set of23

passants (0-lines) of a (q2 + q + 1,2)-arc in the corresponding dual plane. The new arcs24

have intersection numbers 0 and q2/2 with the lines of the dual plane and so are maximal.25

Since Gq = G◦q, the result follows.26

In the smallest case q = 2, the (7,2)-arc in PHG(Z34) is self-dual. In all other cases,27

Example 4.17 gives new arcs not covered by previous constructions, for example a (126,8)-28

arc in the plane over G4.29

?〈thm:dualimp〉?Theorem 4.18. Let R be a chain ring with |R|= 22r, R/N ∼= F2r , charR = 2. Then30

mq2/2(R3R)≤ q4/2−q/2−1. (11)31

Since (q2 +q +1,2)-arcs and((q4−q)/2,q2/2

)-arcs are dual to each other, this theo-32

rem is a corollary of Theorem 4.15.33

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4.6 Constructions using automorphisms1

?〈ssec:singer〉?By the Fundamental Theorem of Projective Hjelmslev Geometry [34], every collineation2

of a coordinate projective Hjelmslev plane Π = PHG(R3R) over a finite chain ring is in-3

duced by a semilinear automorphism of the underlying module R3R, and the collineation4

group of the plane PHG(R3R) is isomorphic to the projective semilinear group PΓL(3,R) =5

ΓL(3,R)/Z(R), where Z(R) denotes the center of the ring R.6

Automorphisms of Π can be used to considerably shorten searches for arcs with good7

parameters and make computer constructions of such arcs feasible which would otherwise8

be out of reach. As a simple example, we mention the fact that one can always assume9

the standard quadrangle (1,0,0)R, (0,1,0)R, (0,0,1)R, (1,1,1)R to be part of K, since10

PGL(3,R) acts regularly on ordered quadrangles in Π.11

The construction of discrete objects using incidence preserving group actions pioneered12

by Kerber et al. [3,25] can also be applied to the construction of arcs in projective Hjelmslev13

planes. To make the resulting computational tasks feasible for larger planes, one restricts14

attention to arcs which are invariant under certain automorphisms of Π, for example (lifted)15

Singer cycles of the factor plane PG(2,q). This method has been used successfully in [18,16

28] for the construction of new arcs with good parameters, accounting for many entries17

(lower bounds) in the tables of Section 4.7. The authors of [28] also maintain online tables18

of optimal arcs in projective Hjelmslev planes of small sizes [1].19

Suppose now that Π is a projective Hjelmslev plane over a Galois ring Gq, represented20

as PHG(Gq3/Gq) (cf. Section 4.4). A generator η of the Teichmüller subgroup Tq3 of G×q321

induces a collineation σ ∈ Aut(Π) of order q2 + q + 1, which acts as a Singer cycle on the22

factor plane PG(2,q). There is obviously a one-to-one correspondence between σ-invariant23

multisets in Π and multisets in a fixed point neighbour class of Π, for example [Gq1]. For24

a σ-invariant multiset K in Π, it is possible to compute the K-types of all lines in Π from25

certain combinatorial data of the corresponding multiset k in [Gq1] ∼= AG(2,q). As shown26

in [17], suitable choices of k yield σ-invariant arcs with good parameters. As an example27

of this construction, we mention a family of arcs in the planes over Gp, where p is an odd28

prime, which includes an optimal (39,5)-arc in the plane over Z9. A multiset k in AG(2, p)29

is called a triangle set if it is affinely equivalent to the set{(x,y) ∈ F2

p | x + y < p− 1}

.30

Here Fp = {0,1, . . . , p−1} is considered as a subset of Z.31

?〈thm:triangleset〉?Theorem 4.19 ( [17]). For every odd prime p, there exists a σ-invariant((p4− p)/2,(p2 +32

p)/2−1)-arc in the projective Hjelmslev plane over the Galois ring Gp. The arc is induced33

from an appropriately chosen triangle set in [Gp1]∼= AG(2, p).34

Finally we want to note that arcs in projective Hjelmslev planes with extremal param-35

eters may be of interest also from a group theoretic point-of-view (just like their classical36

counterparts). This is exemplified by the following result.37

?〈prop:zvenigorod〉?Proposition 4.20 ( [15]). The set H of hyperovals (maximal (7,2)-arcs) of PHG(2,Z4) has38

cardinality 256. The automorphism group G of PHG(2,Z4) acts transitively on H and the39

stabilizer Gh of a hyperoval h ∈ H has order 168. Furthermore, G has a normal subgroup40

H which acts regularly on H.41

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4.7 Tables for arcs in geometries over small chain rings1

〈ssec:tables〉 In the tables below, we summarize our knowledge about the values of mn(R3R) for the chain2

rings R with |R| = q2 ≤ 25, R/ radR ∼= Fq (cf. also [18]). We give information about all3

values of n with 2 ≤ n ≤ q2−1. The cases n = q2, . . . ,q2 +q are covered by Example 4.7.4

We want to remark the fact that we have lots of examples with mn(R3R) 6= mn(S3

S) for non-5

isomorphic chain rings R, S with |R| = |S|, R/ radR ∼= S/ radS (cf. Theorems 4.14, 4.156

and 4.16 and the results in Section 4.5). However, in all these examples charR 6= charS,7

and we do not have a single example of chain rings R and S of the same order, length and8

characteristic, in which the values of mn(R3R) and mn(S3

S) are different.9

n/R Z4 F2[X ]/(X2) Z9 F3[X ]/(X2)2 7 6 9 93 10 10 19 184 30 305 39 386 49 – 51 50 – 517 60 – 62 60 – 628 69 69

Table 1: Values of mn(R3R) for Hjelmslev planes of order q2 = 4 and q2 = 9

n/R G4 S4 T4

0 0 0 01 1 1 12 21 18 183 29 − 30 29 − 30 29 − 304 52 52 525 68 68 686 84 81 − 83 81 − 837 95 − 101 99 − 101 96 − 1018 126 120 − 125 120 − 1259 140 140 14010 152 − 160 152 − 160 152 − 16011 166 − 169 166 − 169 166 − 16912 186 − 189 186 − 189 186 − 18913 201 − 208 202 − 208 202 − 20814 224 − 228 216 − 228 219 − 22815 236 − 248 236 − 248 236 − 248

Table 2: Values of mn(R3R) for Hjelmslev planes of order q2 = 16

?〈tbl:q=4〉?

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n/R Z25 S5

0 0 01 1 12 21 253 40 − 43 42 − 434 66 − 70 64 − 705 85 − 102 90 − 1026 114 − 130 1307 142 − 156 152 − 1568 162 − 186 162 − 1869 186 − 208 190 − 208

10 210 − 238 225 − 23811 234 − 265 250 − 26512 260 − 295 280 − 295

n/R Z25 S5

13 310 − 311 297 − 31114 319 − 341 318 − 34115 355 − 367 355 − 36716 375 − 395 375 − 39517 400 − 425 405 − 42518 425 − 455 433 − 45519 465 − 466 455 − 46620 490 − 496 490 − 49621 515 − 525 515 − 52522 540 − 555 540 − 55523 565 − 585 565 − 58524 595 − 615 595 − 615

Table 3: Values of mn(R3R) for Hjelmslev planes of order q2 = 25

?〈tbl:q=5〉?

5 Blocking sets in projective Hjelmslev planes1

?〈sec:blocking〉?5.1 General results2

?〈ssec:general〉??〈dfn:blockingsets〉?Definition 5.1. A multiset K in (P ,L , I) is called a (k,n)-blocking multiset if3

(i) K(P ) = k;4

(ii) K(L)≥ n for every line L ∈ L .5

Similarly to Definition 4.1, we assume in addition that there exists at least one line L06

with K(L0) = n. A (k,n)-blocking multiset K is called minimal if it does not contain a7

(k−1,n)-blocking multiset, i.e. decreasing the multiplicity of any point p ∈ suppK by one8

yields a multiset K′ with K′(L) = n− 1 for some line L ∈ L . Blocking sets (i.e. projective9

blocking multisets) and projective arcs are complementary concepts in the sense that the10

complement of a projective (k,n)-arc in P is a (q4 + q3 + q2 − k,q2 + q− n)-blocking set11

and vice versa.12

First, let us consider blocking sets in planes over general chain rings R with |R| = qm,13

R/N ∼= Fq. For (k,n)-blocking sets in such planes, we have the following theorem [36].14

〈thm:bsets1〉Theorem 5.2. Let R be a chain ring with |R| = qm, R/N ∼= Fq, and let K be a (k,n)-15

blocking multiset with 1 ≤ n ≤ q, in Π = PHG(R3R). Then k ≥ nqm−1(q + 1). If K is a16

(k,n)-blocking multiset with k = nqm−1(q+1), n < q/p, where p = charFq, then there exist17

lines, L1,L2, . . . ,Ln say, such that18

K(1)([x])= qm−1|{ j | j ∈ {1, . . . ,n},([x], [L j]) ∈ J(1)}|.19

The second part of the theorem says that the induced multiset K(1)/qm−1 is a sum of20

lines. It is impossible to generalize this to the stronger condition: “K(i)/qm−i is a sum of21

lines for some i > 1”.22

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For the most interesting case of (k,1)-blocking sets, we have k ≥ qm−1(q + 1) and1

in case of equality the support of such a blocking set is necessarily a line. By taking a2

line L and from each class [x]m−1 incident with [L]m−1 in (P (m−1),L(m−1),J(m−1)) exactly3

n− 1 further line segments in the direction of L, one obtains for each n ∈ {1,2, . . . ,q} an4 (n,nqm−1(q + 1)

)-blocking set, showing that the extremal cases k = nqm−1(q + 1) of The-5

orem 5.2 can be realized by projective multisets.6

Under certain conditions, some subplanes of PHG(R3R) form a blocking set.7

?〈thm:subplane〉?Theorem 5.3. Let R be a chain ring with |R|= qm, R/N ∼= Fq, where qm is a perfect square.8

Let there exist a subring S of R that is a chain ring with |S| = qm/2 and such that R is free9

over S. Then the multiset K defined by10

K(x) ={

1 if x is a point from PHG(S3S),

0 otherwise,11

is a blocking set in PHG(R3R).12

In the special case when R is a chain ring with |R| = q2, R/N ∼= Fq, that contains a13

subring S isomorphic to the residue field Fq, PHG(R3R) contains a subplane Π′ isomorphic14

to PG(2,q) and the projective multiset K defined by suppK = Π′ is an irreducible (q2 +q+15

1,1)-blocking set. These blocking sets are introduced in [5] in a slightly different context.16

They are defined as the orbit of a fixed point with coordinates from the field Fq under a17

Singer cycle of PG(2,q). As shown in [6], linear codes associated with these multisets can18

be mapped (cf. [20]) to two-weight linear codes over Fq. These in turn give rise to a family19

of strongly regular graphs with parameters20

v = q6, k = q4−q, λ = q3 +q2−3q, µ = q2−q.21

Let us now consider planes over chain rings with |R| = q2, R/N ∼= Fq. It is of interest22

to find the smallest size of a minimal blocking set which is not a line. Unlike the situation23

in the classical projective planes where there is a gap between the size of a line and the size24

of the smallest non-trivial blocking sets (see e.g. [4]), there exist minimal blocking sets of25

size q2 +q+1 in all planes PHG(R3R).26

?〈thm:nonexbaer2〉?Theorem 5.4. Let K be a minimal (q2 + q + 1,1)-blocking set in PHG(R3R), |R| = q2,27

R/ radR ∼= Fq. Then K is of one of the following types:28

(1) a projective plane of order q;29

(2) for lines L0 and L1 with L0 _ L1, and a point z ∈ L0 \L130

K(x) ={

1 if x ∈ (L0 \ [z])∪{z} or x ∈ L1∩ [z]0 otherwise.

(12)31

If R = GR(q2, p2), then there is no (q2 +q+1,1)-blocking set of type (1).32

Let us note that the blocking set described in (12) is in some sense trivial since K(1) =33

q · χ[L] + χ[z] consists of a q-fold line and a further point on this line. We would like to34

construct non-trivial blocking sets also for the planes over the Galois rings Gq. This can35

be done by generalizing the familiar technique of Rédei type blocking sets to projective36

Hjelmslev planes.37

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5.2 Rédei type blocking sets1

?〈ssec:redei〉?As before let Π = PHG(R3R), where R is a chain ring of nilpotency index 2. Fix a generator θ2

of radR and a set Γ⊂ R of representatives for the residue classes in R/ radR∼= Fq. Suppose3

that Γ = {γ0,γ1, . . . ,γq−1} with γ0 = 0, γ1 = 1, and hence radR = {γiθ | 0 ≤ i ≤ q− 1} =4

{θγ j | 0≤ j ≤ q−1}. Thus each c ∈ radR has unique representations c = γiθ = θγ j, where5

in the non-commutative cases i, j may be different.6

As already noted, the affine plane AHG(R2R) is obtained by deleting a neighbour class7

of lines (the “class at infinity”) together with all points incident with a line in this class.8

With no loss of generality we can take the class [z = 0] as the class at infinity. This class9

consists of all lines with equations of the form aX + bY + Z = 0, where a,b ∈ radR. All10

points incident with lines in this class have homogeneous coordinates (x,y,z) with z∈ radR.11

The points outside this class have coordinates (x,y,1), x,y ∈ R. Now the points of the affine12

plane AHG(R2R) are identified with the pairs (x,y), where x,y ∈ R. The lines of AHG(R2

R)13

have equations Y = aX + b or X = cY + d, a,b,d ∈ R, c ∈ radR. We say that a line of the14

first type has slope a. A line with equation X = cY +d is said to have slope ∞ j, if c = θγ j,15

j = 0,1, . . . ,q−1.16

The infinite points on a fixed line L from the neighbour class of infinite lines can be17

identified with the slopes. So, (a) (resp. (∞ j)) will denote the infinite point from L of the18

lines with slope a (resp. ∞ j). The q2 lines with a fixed slope form a parallel class of lines19

in AHG(R2R), and the line set of AHG(R2

R) is partitioned into q2 +q such parallel classes.20

Definition 5.5. Let U be a set of q2 points in AHG(R2R). We say that the infinite point (a)21

is determined by U if there exist different points u,v ∈U such that u,v and (a) are collinear22

in PHG(R3R).23

Note that in view of the assumption |U |= q2, the point (a) is determined by U iff there24

exists a line in AHG(R2R) with slope a which is disjoint from U .25

?〈thm:main〉?Theorem 5.6. Let U be a set of q2 points in AHG(R2R). Denote by D the set of infinite points26

determined by U and by D(1) the set of neighbour classes on the infinite line containing27

points from D. If |D|< q2 +q, then there exists a minimal blocking set in PHG(R3R) of size28

q2 + q + 1 + |D|− |D(1)| that contains U. In particular, if D contains representatives from29

all neighbour classes on the infinite line, then B = U ∪D is a minimal blocking set of size30

q2 + |D| in PHG(R3R).31

The above construction gives blocking sets of size at most 2q2 +q−1. We are interested32

in sets U that are of the form33

U = {(x, f (x)) | x ∈ R}34

for some suitably chosen function f : R→ R. Let x and y be two different elements from R.35

We have the following possibilities:36

1) if x− y 6∈ radR, then (x, f (x)) and (y, f (y)) determine the point (a), where37

a = ( f (x)− f (y))(x− y)−1.38

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2) if x−y∈ radR\{0}, and f (x)− f (y) 6∈ radR, the points (x, f (x)) and (y, f (y)) determine1

the point (∞ j) if2

(x− y)( f (x)− f (y))−1 = θγ j, γ j ∈ Γ.3

3) if x−y ∈ radR\{0}, and f (x)− f (y) ∈ radR, say x−y = θa, f (x)− f (y) = θb, a,b ∈ Γ4

and5

a) if b 6= 0, then (x, f (x)) and (y, f (y)) determine all points (c) with c ∈ ab−1 + radR;6

b) if b = 0, then (x, f (x)) and (y, f (y)) determine the infinite points (∞0), . . . ,(∞q−1).7

Example 5.7. Let R be a chain ring with |R| = q2, R/ radR ∼= Fq that contains a proper8

subring isomorphic to its residue field Fq (i.e. one of the rings Sq or T(i)q ).9

Define10

f : R → R : a+θb 7→ b+θa. (13)11

It can be checked that the set of points U = {(x, f (x)) | x ∈ R} determines q + 1 infinite12

points.13

We can compute the parameters of the Rédei-type blocking sets given by (13) also for14

the plane over the Galois ring Gq = GR(q2, p2). In this case, U determines exactly q2−q+215

directions, and the size of the corresponding Rédei-type blocking set is 2q2−q+2.16

Below we will give two further of examples Rédei-type blocking sets in the plane over17

Gq. For these examples, we need to collect a few additional facts about Gq.18

In the case of Gq (and Galois rings in general), there are canonical choices for θ and19

Γ, which we will adopt for the rest of this paper. Since radGq = pGq, we can take θ = p20

as a generator of radGq. Furthermore, since the augmented Teichmüller subgroup Γq :=21

Tq∪{0} (for the definition of Tq, see Section 4.4) forms a system of coset representatives22

modulo radGq, we can take Γ = Γq.23

Every a ∈Gq can be written in exactly one way as a = a0 +a1 p with a0,a1 ∈ Γq.524

〈fact:witt〉Fact 5.8. The ring Gq is isomorphic to the ring W2(Fq) of so-called Witt vectors of length2 over Fq, which is defined as the set of all pairs (a,b)∈ Fq×Fq with the following additionand multiplication:

(a0,a1)+(b0,b1) = (a0 +b0,a1 +b1−p−1

∑j=1

1p

(pj

)a j

0bp− j0 ),

(a0,a1) · (b0,b1) = (a0b0,ap0b1 +bp

0a1).

The map φ : Gq →W2(Fq) : a0 + a1 p 7→ (a0,a1p), where a = a + radGq, provides a ring25

isomorphism.26

For the definition of Witt vectors see [45], and for a proof of 5.8 see [43]. Working with27

Witt vectors instead of the original representation of Gq = Zp2 [X ]/(h) has the advantage28

that all computations are now done in Fq.29

5This is true regardless of the particular choice of Γ. However, for the following Fact 5.8 the choice Γ =Tq∪{0} is essential.

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Example 5.9. Let q = pr, where p is odd. We are going to define f as a function on W2(Fq).1

For x = (a0,a1), set2

f (x) ={

(a0,a1) if a0 is a square in Fq,(−a0,−a1) if a0 is a non-square in Fq.

(14)3

?〈thm:square〉?Theorem 5.10. Let R = GR(q2, p2), q = pm, p odd. The set U = {(x, f (x)) | x ∈ R}, where4

the function is defined in (14), determines5

q2

2+

32

q6

directions in AHG(R2R). Furthermore, there exists a Rédei type blocking set in PHG(R3

R) of7

size8

32

q2 +2q− 12.9

In our last example, we will construct a Rédei type blocking set over the Galois ring10

S = Gqm , where m ≥ 1 is arbitrary, using the fact that S is a Galois extension of R = Gq.11

Recall that the trace function TrS/R : S → R is defined as12

TrS/R(x) := ∑σ∈Aut(S/R)

σ(x) =m−1

∑i=0

(xqi

0 + xqi

1 p) for x ∈ S, (15)13

where x = x0 + x1 p with x0,x1 ∈ Γqm .14

Example 5.11. As above let R = Gq and S = Gqm . We define a Rédei type blocking set in15

PHG(S3S) by setting f (x) = TrS/R(x).16

?〈thm:trace〉?Theorem 5.12. Let R = GR(q2, p2) and let S be an extension of R of degree m. The set17

U = {(x, f (x)) | x ∈ S} defined by the function f (x) = TrS:R(x) determines18

qm−1q−1

qm19

directions in AHG(S2S). There exists a Rédei type blocking set in PHG(S3

S) of size20

q2m +qm +1+qm−1q−1

qm−qm−1.21

Acknowledgement22

The authors wish to thank Michael Kiermaier for help with the tables in Section 4.7 and23

with Examples 4.10 and 4.17.24

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