On the sizes of expander graphs and minimum distances ofgraph codesTom Høholdt ∗, Jørn JustesenDepartment of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet, Building 303B,DK-2800 Kgs. Lyngby, Denmark
a r t i c l e i n f o
Article history:Received 5 October 2012Received in revised form 5 February 2014Accepted 7 February 2014Available online 5 March 2014
Keywords:Graph codesMinimum distanceEigenvalue
a b s t r a c t
We give lower bounds for the minimum distances of graph codes based on expandergraphs. The bounds depend only on the second eigenvalue of the graph and the parametersof the component codes. We also give an upper bound on the size of a degree regular graphwith given second eigenvalue.
We consider n-regular connected bipartite graphs withm nodes on each side. The graph may be described by anm bymtransfer matrixM , or alternatively by a symmetric 2m by 2m adjacency matrix,
A =
0 MMT 0
.
Here MT denotes the transpose of the matrix M . Similarly we consider non-bipartite graphs with symmetric adjacencymatrix A. Let
λ1 ≥ λ2 ≥ · · · ≥ λ2m
be the eigenvalues of A.The largest eigenvalue is n, and we are interested in graphs where the second largest eigenvalue λ = λ2 is small. This
comes from the fact that such graphs are good expanders (see [14]) and that they also give graph codes with goodminimumdistance (see below). A classical result by Alon and Boppana [13] is
Theorem 1. For every fixed positive integer n,
lim infG
λ ≥ 2√n − 1
where the limit is taken over any infinite sequence of distinct n-regular graphs G .
T. Høholdt, J. Justesen / Discrete Mathematics 325 (2014) 38–46 39
It is well known [4] that there are arbitrarily large graphs such that λ = 2√n − 1.
Instead of working with thematrix Awewill sometimes work with the Laplace matrix Q = nI −A. Since the eigenvaluesλ(A) of A are connected to the eigenvalues λ(Q ) of Q by
λ(Q ) = n − λ(A)
a small eigenvalue of A corresponds to a large eigenvalue of Q .In this paper we will derive upper bounds on the size of a graph when λ is smaller than 2
√n − 1.
By a similar argument we derive a lower bound on the minimum distance, D, of a graph code with component (n, k, d)codes (on both sides) based on a bipartite graph with a given value of λ. This bound generalizes the bound for bipartitegraphs [1, Theorem 5]
D ≥ dmd − λ
n − λ.
In the non-bipartite case we have
D ≥ dm′d − λ
2(n − λ)
where m′ is the number of nodes.The aim of the graph code construction is to find very long codes from short to moderate component codes. The over-all
length is directly related to the size of the graph. Sincewewant to keep n and (in particular) d relatively small, it is importantthat λ is small. It is therefore of interest to make the graph as large as possible for a given eigenvalue (or to minimize theeigenvalue for a graph of given size). We start out by deriving upper bounds on the sizes of graphs in Sections 2–3. We thenshow that the same technique can be used to prove lower bounds on the minimum distances of the corresponding graphcodes. This extends the result obtained in [8].
2. Upper bounds on the sizes of graphs with given second eigenvalues
In the following we make use of the quotient matrix of a (symmetric) matrix A, so for the sake of completeness we recallthe definition from [6]. Let A be an N ×N matrix. Suppose rows and columns of A are partitioned according to a partitioningX1, . . . , XM of {1, . . . ,N} with characteristic matrix S of size N × M (that is (S)i,j = 1 if i ∈ Xj and 0 otherwise), so
A =
A1,1 · · · A1,M· ·
· ·
· ·
AM,1 · · · AM,M
.
The quotient matrix is the matrix B of sizeM × M whose entries are average row sums of the blocks of A, more precisely:
(B)i,j =1
|Xi|1TAi,j1 =
1|Xi|
(STAS)i,j
(1 denotes the all-one vector). The partition is called regular if each block Ai,j of A has constant row (and column) sum, thatis AS = SB.
Theorem 2 ([6]). Suppose B is the quotient matrix of a symmetric partitioned matrix A.
(i) The eigenvalues of B interlace the eigenvalues of A.(ii) The interlacing is tight if and only if the partition is regular.
If λ1, λ2, . . . , λN are the eigenvalues of A and µ1, µ2, . . . , µM are the eigenvalues of B the first statement means that
λi ≥ µi ≥ λN−M+i, for i = 1, . . . ,M
and the interlacing is called tight if there exists an integer k ∈ [0,M] such that
λi = µi for 0 ≤ i ≤ k andλN−M+i = µi for k + 1 ≤ i ≤ M.
We first consider the case of connected bipartite graphs with degree n on both sides. The nodes will be partitioned intosubsets in the following way: initially choose a set consisting of two connected nodes (obviously one on each side). Thencollect all nodes at distance j from this pair into a subset for j = 1, 2, . . . . The maximum number of nodes that can bereached in j steps from an initial node is
M(j) = 1 + (n − 1) + (n − 1)2 + · · · + (n − 1)j.
40 T. Høholdt, J. Justesen / Discrete Mathematics 325 (2014) 38–46
Clearly this number is reachedwhen each node at distance l is reached from a single node at distance l−1, and n−1 distinctnodes at distance l + 1 are reached from such a node. In this case we find the quotient matrix
of size j + 1. The eigenvalues are n, and pairs of values with opposite signs, 2
√n − 1 cos( kπ
j+1 ), k = 1, 2, . . . . This can beseen by applying the same method that is used in the solution to problem 1.29 in [11]. If j is odd, 0 is an eigenvalue withmultiplicity 1.
Theorem 3. Let G be an n-regular connected bipartite graph with m nodes on each side and A the adjacency matrix. If m ≥ M(j)then the second largest eigenvalue λ of A satisfies
λ ≥ 2√n − 1 cos
π
j + 1
.
Another way of stating the theorem is that if λ ≤ 2√n − 1 cos( π
j+1 ) thenm ≤ M(j). We note that the bound can be reachedonly if the graph has girth 2j+ 2. The theorem is new and we will prove it in Section 6. In [10] the authors using orthogonalpolynomials show that λ ≥ 2
√n − 1 cos( 2π
g ) if the girth g is even and a similar result for odd g .Clearly it is possible to construct a matrix of size M(j) where the nodes are reached in j steps from a particular node,
but it may not be possible to have a matrix where this structure is found for every initial pair of nodes. In fact it is knownthat only the cases j = 1, 2, 3, and 5 exist [5]. For j = 3 and 5, the corresponding graphs are generalized quadrangles andhexagons. For large j the eigenvalue approaches 2
√n − 1.
Thus we have provided lower bounds for the second largest eigenvalue for allm. In the following section we will discussimprovements forM(j) ≤ m ≤ M(j + 1).
Graph codes are most often constructed from bipartite graphs, but codes on non-bipartite graphs were presented asearly as in [15]. Actually the only advantage of bipartite graphs appears to be that they suggest a natural iterative decodingschedule alternating between the two sides of the graph. However, recent applications suggest that a single set of componentcodes may be preferable for some implementations [9]. For a general connected graph with degree n, the bound on theminimum distance depends on the second largest eigenvalue (which may not be numerically the second largest [3]).
The analysis of a general n-regular connected graph proceeds in a way very similar to the argument for bipartite graphswith the exception that the initial set consists of a single node. Thus the maximum number of nodes that can be reachedfrom the initial set in j steps becomes
The matrix B′ is modified to B′′, which has the first column start (0, n, 0, . . .) since the initial node is connected to n othernodes. By the same argument as before, M ′(j), becomes an upper bound on the size of a matrix with the second eigenvaluegiven by the corresponding value in B′′.
For a general graph, the bound is reachedwith equality form = M ′(2) = n2+1 in the case of so-calledMoore graphs [12].
It is known that they exist only in the cases n = 2, 3, 7, and possibly 57. The second eigenvalue is√n − 3/4 − 1/2. For
higher values of j, these graphs would be N-gons with odd values of N , but no such structures exist [5].
3. Upper bounds onm for special values of λ
As noted elsewhere [7], the second eigenvalue for an n-regular connected bipartite graph with m nodes on each side islower bounded by
λ ≥
(mn − n2)/(m − 1).
The result follows from observing that for the adjacency matrix A, Tr(AAT ) is twice the number of edges and also the sum ofthe squares of all eigenvalues in A.
In this section we shall derive lower bounds on the second eigenvalue of A for several intervals of values for m bypartitioning A and applying the interlacing theorem. First we partition the nodes into 6 sets: A′
0 and A′′
0 are two singleconnected nodes, A′
1 and A′′
1 are the sets of n − 1 nodes on each side that can be reached from one of these nodes, andA′
2 and A′′
2 are the remainingm − n nodes on each side. This gives the quotient matrix
Ba =
0 B1B2 0
T. Høholdt, J. Justesen / Discrete Mathematics 325 (2014) 38–46 41
but since the number of edges from A′
i to A′′
j is the same as from A′′
i to A′
j , we have B1 = B2 = B, where
B =
1 1 0n − 1 x y0 n − 1 − x n − y
.
Counting the number of edges between A′
1 and A′′
2 in two ways gives
y(m − n) = (n − 1)(n − 1 − x).
It is easy to see that the eigenvalues of Ba are ± the eigenvalues of B. Thus if the eigenvalues of B are n, µ1, µ2, then λ islower bounded by max{|µ1|, |µ2|}, and we get the minimal values when µ1 = γ and µ2 = −γ . In this case
x = n(n − 1)/(m − 1) − 1
and we get
γ =
(mn − n2)/(m − 1).
By applying the interlacing theoremwe then find that the second eigenvalue of A is lower bounded by γ . This bound appliesfor
n ≤ m ≤ n2− n + 1
where λ goes from 0 to√n − 1. The largest graph for which equality holds is the incidence graph for the projective plane.
Similarly, for graphs of sizes between 1 + n + n2 and 1 + n + n2+ n3 we get the quotient matrix
B =
1 1 0 0n − 1 0 1 00 n − 1 x y0 0 n − 1 − x n − y
.
The number of nodes on each side in the subsets are 1, n − 1, (n − 1)2, and r . Counting the number of edges between twosets of the partition in two ways we have
(n − 1)2(n − 1 − x) = ry.
By varying y to get the smallest second eigenvalue for a given r , we find y = x + 1, which gives the eigenvalues±
√2(n − 1) − x. As x goes from n − 1 to 0, this eigenvalue goes from
√n − 1 to
√2(n − 1). The size of the last subset
is
r = (n − 1)2(n − 1 − x)/(x + 1)
which goes from 0 to (n − 1)3.
Theorem 4. A bipartite graph of degree n with second eigenvalue λ,√n − 1 ≤ λ ≤
2(n − 1)
has m nodes on each side, where m is upper bounded by
m ≤ n +n(n − 1)2
2n − λ2 − 1.
In Section 6 we prove that the minimum for the second eigenvalue in the quotient matrix is a lower bound on theminimumfor the actual matrix.
Similar expression can be found for the following intervals between the special values of the second eigenvalue.Euclidean planes give bipartite graphs of degree q and size q2 (on each side). The bound in this section gives λ =
√q for
x = q − 2. Thus the second eigenvalue is minimal for the given degree and size (and the multiplicity of the zero eigenvaluecannot be reduced to give a smaller value of λ).
For a general graph of size 2n < m < n2+ 1, we get then smallest (positive) second eigenvalue in the quotient matrix
for
B =
0 1 0n 0 y0 n − 1 n − y
.
The value of y follows from the condition that the eigenvector for eigenvalue n is (1, n,m − n − 1).For the bounds to be met with equality it is necessary and sufficient that the graph is distance regular [2], in which case
the matrix B becomes the intersection matrix of the graph.When N-gons exist, they have minimum diameter and maximum girth for graphs of the given size. It is demonstrated
in [12] that there are graphs which come quite close to the Moore bound. We expect the same to be true for graphs withgiven second eigenvalues.
42 T. Høholdt, J. Justesen / Discrete Mathematics 325 (2014) 38–46
4. Improved lower bounds on the minimum distance
We recall the definition of a graph code.Let G = (V , E) be an n-regular connected bipartite graph, without multiple edges, with vertex set V = V1 ∪ V2 such
that V1 ∩ V2 = ∅ and |V1| = |V2| = m. Let C1 be a linear (n, k1, d1) code and C2 a linear (n, k2, d2) code both over thefinite field Fq. We now construct a code C of length L = mn over Fq by associating Fq symbols with the edges of the graphand demanding that the symbols connected to a vertex of V1 shall be a codeword of C1 and that the symbols on the edgesconnected to a vertex of V2 shall be a codeword of C2. More formally we assume an ordering of the edges E of G and for avertex u ∈ V let E(u) denote the set of edges incident with u. For a word x = (xe)e∈E in Fq
L denote by (x)E(u) the subword ofx that is indexed by E(u), that is (x)E(u) = (xe)e∈E(u). Then the code C is defined by
C = {c ∈ FqL: (c)E(u) ∈ C1 for every u ∈ V1 and (c)E(u) ∈ C2 for every u ∈ V2}.
It is clear that C is a linear code. In the following we consider the case where C1 = C2 and therefore k1 = k2 = k andd1 = d2 = d and call this code the component code. In a bipartite graph, a minimum weight codeword has a nonzerocomponent word associated with a set of nodes of size a on one side and ≥a on the other side of the graph. Clearly thissubgraph has degree at least d. If there are more than a nodes on one side, we remove the ones of lowest weight to obtaina subgraph of size 2a with at least da edges (it may then no longer support a codeword). The remaining edges from thesenodes can reach at most 2a(n− d) other nodes. As in the previous sections, the following argument is based on dividing thenodes of the graph into subsets such that the nodes in each subset have a certain distance from an initial subset of nodes.But we now take the nonzero set of 2a nodes as the initial set. We can get a lower bound on the minimum distance of graphcodes by makingm as large as possible for a given second eigenvalue of the quotient matrix and a given a. As before we usethe interlacing theorem to get lower bounds on the second eigenvalue of the adjacency matrix.
Dividing the nodes of the graph into two subsets, the 2a nodes as above and the remaining nodes, we get the quotientmatrix
C =
d x
n − d n − x
with eigenvalues n and d − x. Since for the eigenvalue n the eigenvector is [a,m − a], we get a =
xmn−x−d and from the
interlacing theorem we have λ ≥ d − x and hence x ≥ d − λ and therefore a ≥m(d−λ)
n−λand since D ≥ da this gives the
bound on the minimum distance of the graph code mentioned in the introduction:
D ≥ dmd − λ
n − λ.
It follows from the results in the previous sections that for λ < 2√n − 1 there is an upper bound on m for given n and
λ. Clearly the bound can be reached only for d ≥ λ + 1, since x ≥ 1. For smaller values of d, we divide the set of nodes intosubsets according to their distance from the initial set.
The largest number of nodes that can be reached in j steps from the initial set is
As in B′ considered earlier, the eigenvalues are n, pairs of values with opposite sign, and possibly 0. The eigenvalues canbe calculated explicitly by standard methods, but there is not a simple expression as in the case of B′. The second eigenvaluewas given earlier for d = 1, and it increases with increasing d.
Theorem 5. For a graph code on a bipartite graph of size m, degree n, and second eigenvalue λ, and for a givenminimum distanced of the component code between 1 and 1 + λ, let j be the size of the smallest C ′
j such that the second eigenvalue is ≥λ. Then theminimum distance of the graph code is lower bounded by
D ≥ ad
where a = m/M ′(j).
T. Høholdt, J. Justesen / Discrete Mathematics 325 (2014) 38–46 43
The theorem is proved in Section 6.The theorem can be applied to any permissible combination ofm and λ and any value of d ≤ λ + 1. For larger dwe have
given a simple bound previously.It may be of interest to note how small d/n can be while the lower bound on D/N is still positive for arbitrarily large
graphs. For large graphs it is sufficient to consider λ = 2√n − 1. To get a positive value of a/m, C ′′
j must have this secondeigenvalue for a finite j. For this value of λ = α/2, the coordinates of the eigenvector for the second eigenvalue satisfy arecursion with the double root α. Since the vector should be balanced, it must have the form αi
+ kiαi with some constantk < 0. The limit k = 0 is reached for d = α + 1. Thus we get a lower bound D/N > 0 for d > 1 +
√n − 1. Of course there
can be graph codes with positive relative distance and component codes with smaller d/n, but in that case the mapping ofthe symbols to the branches of the graph have to be chosen in a special way or randomized.
Since quotient matrices of small sizes are most useful, we consider j = 3 and 4 in detail, and get sharper bounds in thesecases. For 3 subsets we get
C3 =
d 1 0n − d n − g − 1 h0 g n − h
.
The largest eigenvalue of C ′
3 is n, and the corresponding eigenvector is (1, n− d, f ). We want to choose g ≤ n− 1 and h ≤ nin such a way that f is maximized while the largest numerical value of the remaining two eigenvalues is λ. Alternatively wecan fix f by taking a particular value of g/h andminimize λ. Solving for λ, we can find a localminimumwhen the eigenvaluesare ±λ. In this case we have
C3 =
d 1 0n − d x d + x0 n − x − 1 n − d − x
.
We find
λ2= n + d2 − 2d − x
which is valid from x = n − 1 to 0 or
λ + 1 ≥ d ≥ 1 +
λ2 − n + 1.
Theorem 6. For a graph code with component (n, k, d) code on a bipartite graph of degree n with second eigenvalue λ,
λ + 1 ≥ d ≥ 1 +
λ2 − n + 1
and m nodes on each side, the minimum distance is upper bounded by D ≥ ad, where
am
≥n + d2 − d − λ2
n2 − λ2.
Proof. Since (1, n − d, f ) is an eigenvector for the eigenvalue n we get f =(n−1−x)(n−d)
d+x and by construction we havem ≤ a(1 + (n − d) + f ) we get
am
≥1
1 + n − d + f.
Inserting x = n + d2 − 2d − λ2 gives the inequality above and the result follows.
For x > n − d the last entry in C becomes negative, and we get a tighter bound in the range λ + 1 ≥ d ≥ λ + 1/2 bytaking h = n. In this case we find
ma
= 1 + n − d +(n − d)(n − x − 1)
nx =
λ2
+ λ(n − d) −d(n − 1)λ − d
.
Similarly for d < 1 +√
λ2 − n + 1, we get the maximum on the boundary g = n − 1 and
h =λ(d − λ) + n − d
λ − d − 1
which is valid for h ≥ 1 or d ≥ λ −n−1λ+1 .
For the particular case d = λ we get the simple bound D ≥mdn+d . For d = 1 +
√λ2 − n + 1, the bound becomes
D ≥md2
n2−λ2. For even smaller values of d, the nodes have to be split into 4 sets, and we again find three intervals for d, a case
44 T. Høholdt, J. Justesen / Discrete Mathematics 325 (2014) 38–46
on the boundary where the last column is [0, 0, n, 0], a range with two eigenvalues of the same numerical magnitude (andan additional, usually nonzero eigenvalue), and a case on the boundarywhere the second to the last column is [0, 1, 0, n−1].
The easiest point to calculate is the lower end of the second interval, which corresponds to a matrix with the last columnending in d, n − d. This case gives the matrices C ′
j considered above.For a general n-regular graph we similarly split the set of vertices into an initial set of a nodes and two other sets. The
same derivation gives lower bounds on the minimum distance that are half the value of result for bipartite graphs (the totallength is alsomn/2 compared tomn).
As proved in [6], for the graph to have a subgraph of degree d and size a it is necessary and sufficient that the distancepartitioning is regular, in which case the larger eigenvalues of A are also eigenvalues of the quotient matrix.
5. Applications to some classes of graph codes
In this section we demonstrate that the lower bound on the size of a subgraph with degree d is reached with equalityin several cases. These examples also provide some additional insight into the geometric significance of the bounds. Evenwhen such a subgraph exists, it may not support a nonzero codeword ofminimumweight. For a binary component code, thecodemay not have aminimumweight codeword on those particular positions, whereas for a Reed–Solomon code, althougha minimum weight codeword exists, it may not be possible to find nonzero values on the edges that satisfy all constraints.On the other hand, if we allow generalized RS codes as component codes, it is always possible to choose multipliers for thecode positions in such a way that the subgraph supports a minimumweight codeword. The actual minimum distance of thegraph code does not necessarily determine the number of errors corrected, since the code is most effectively decoded byiterating a decoding algorithm for the component codes. However, a large minimum distance is desirable in order to ensurea small probability of decoding error [9].
The lower bound on the second eigenvalue is reached by Moore graphs, but as mentioned in Section 2, only a few suchgraphs exist. We provide the following example of a graph code based on such a graph.
Example 1. Codes on the Moore graph with n = 7. The (Hoffman–Singleton) graph has m = 50 nodes, and the secondeigenvalue is 2. Let the minimum distance of the component code be 3. Thus we get sets of size a and 4a, a = 10. Theminimum distance of the code is D ≥ 3 · 10/2 = 15. A subgraph of nonzero edges with degree 3 on 10 nodes would be aMoore graph, in this case the Petersen graph.
The remaining codes considered in this section are based on bipartite graphs. Initiallywe note that graph codeswith sizesm between product codes wherem = n and codes on planes wherem = n2
− n+ 1 can be derived from various designs. Inapplications product codes are widely used, and since the component codes are rather long, it may not be practical at thispoint to use codes that are much longer [9]. An adjacency matrix can be obtained by complementing the connection matrixof the projective plane, thus the component code has length n = q, and the size becomes m = n +
√n + 1. In this case
λ =√q = n1/4. The distance of the graph code is greater than d2 for sufficiently large d. Thus the lower bound is better than
for product codes even for a slightly longer code.
Example 2. Short graph code. For a bipartite graph of size 273 and component codes of length 256, the lower bound on theminimum distance becomes
D ≥ md(d − 4)/252
which exceeds d2 for d > 52.
When the graph is the incidence matrix of points and lines in a projective plane over Fq, it has q2 + q + 1 nodes on eachside, and the component code has length n = q + 1. As noted earlier the second eigenvalue is
√q. If q is a square, we can
choose d =√q+1, and the bound from Section 1 gives D ≥ d(q2 +q+1)/(q−
√q+1) = d/(q+
√q+1). Thus in this case
the subgraph is the incidence graph of a subfield plane over F√q. The projective plane can be partitioned into q −
√q + 1
disjoint copies of the subplane, and each line in one of the subplanes has one point in each of the other copies. In this waywe can find a subgraph of degree
√q+ s by taking the union of s copies of the subplane. In each case we get a subgraph that
satisfy the lower bound with equality.
Example 3. Codes on projective planes. Consider the projective plane over F16. Here there are 273 nodes on each side, andthe component code has length n = 17. Thus for d = 5 we can have a minimum codeword supported by a subfield plane
D ≥ md(d − 4)/(n − 4) = 273 ∗ 5/13 = 85
whereas for d = 6 we get D ≥ 6 ∗ 2 ∗ 21 = 252.
For codes based on the Euclidean plane, a subfield plane may not give a minimum weight codeword. In F16 there are252 nodes on each side, and the second eigenvalue is 4 as for the projective plane. For a component (16, k, 4) code we geta ≥ 256/(16+ 4). A graph of degree 4 with 13 nodes could be found as the incidence graph of the projective plane over F3,but it is not a subgraph of the incidence graph for the Euclidean plane. However, we can partition the plane into q disjoint
T. Høholdt, J. Justesen / Discrete Mathematics 325 (2014) 38–46 45
copies of the subfield plane in such a way that a line in the subplane intersects q −√q other copies in one point. We can
then get codes with larger distances by taking the union of several such copies.
Example 4. Codes on a Euclidean plane. For q = 16 we can then take a union of 4 subplanes to get a subgraph of degree7 and size 64. This subgraph satisfies the lower bound, and we can similarly get a graph of degree 10 and size 128 on thelower bound by taking the union of 8 subplanes.
The generalized quadrangle constructed from projective space over Fq has q3 + q2 + q + 1 lines and points. The secondeigenvalue is
√2q. In general, for q a square, d =
√q+ 1 is the endpoint of the interval for d in the previous section, and we
get D = md2/(q2 + 1). If the quadrangle is restricted to Euclidean space, the second eigenvalue is still√2q, but this value is
a little larger than the bound in Section 3. It is not known whether there are larger graphs with this eigenvalue. Using thisgraph we again consider quadrangles on a subfield and form unions of several such sets.
Example 5. Codes on generalized quadrangles. The generalized quadrangle constructed from projective space over F8 has585 nodes on each side. The second eigenvalue is 4. For a component (9, k, 4) code (since the alphabet is not important, wecan think of an extended RS code), the derivation above immediately gives D ≥ 585 · 4/13 = 180. If the graph is chosenas the generalized quadrangle over F16, the second eigenvalue is 4
√2. For d = 5 the bound of this section gives a ≥ 85.
Thus in this case the F4 quadrangle can support a minimum weight codeword. For d = 6 we get a ≥ 255, which could bemet with a union of 3 subfield quadrangles. If the quadrangle is restricted to Euclidean space with q = 16 and d = 6, theprevious section gives a ≥ 256. Since the subfield plane has 64 nodes, this boundmay be met by a union of 4 disjoint copiesof the subplane where a line in one subplane intersects 2 of the other copies in 1 point.
To extend the lower bound to smaller values of d (compared to λ), we need to separate the nodes into 4 or more subsets.We omit the details, but provide the following example:
Example 6. Codes on a generalized hexagon. The generalized hexagon over F16 has second eigenvalues 4√3. Thus the
original lower bound applies for d ≥ 8. For d = 7 we get a lower bound from Theorem 2, a ≥ 51 051. For d = 5 weare at the lower end of the second interval for 4 subsets, and we get the bound a ≥ 1365. Thus in this case the F4 hexagoncan support a minimum weight codeword.
6. Proof of theorems
In this section we first prove Theorem 3, and we indicate how the proof should be modified to be applied to the othertheorems.
Let A be the adjacency matrix of an n-regular connected bipartite graph Gwith m nodes on each side where
m = M(j) + r, 0 ≤ r < (n − 1)j+1.
Let A0 be a set of two connected nodes and Ai be the set of nodes at distance i from A0, i = 1, . . . , j + 1. We partitionthe nodes of G in the following way: B0 = A0; if s is the smallest integer such that |A1 ∪ A2 ∪ . . . As| ≥ 2(n − 1), letB1 = A1 ∪ · · · ∪ As−1 ∪ R where R ⊆ As such that |B1| = 2(n − 1). For 1 < i ≤ j, let V be the set of nodes connected to Bi−1that are not in any of the Bl for l < i − 1 and if t is the largest integer such that V ∩ At = ∅ and t1 the smallest integer suchthat |V ∪ At ∪ · · · ∪ At+t1 | ≥ 2(n− 1)i then Bi = V ∪ At ∪ · · · ∪ At+t−1 ∪ V ′ where V ′
⊆ At+t1 such that |Bi| = 2(n− 1)i. LetBj+1 be the remaining set of vertices, |Bj+1| = r , and let z be the number of edges from Bj to Bj+1. Let B be the (j+2)× (j+2)matrix
B =
0 0
B′′...
...1 0x y
0 · · · 0 n − 1 − x n − y
where B′′ is the submatrix of B′ consisting of the first j columns. Here y = z/r to give the right number of branches betweenthe last subsets, and by counting these branches we get as in Section 3 (n− 1)j(n− 1− x) = ry. This holds if r > 0. If r = 0we have x = n − 1 and y = n or undefined.
Let u = (1, u1, . . . , uj+1) be an eigenvector of B corresponding to its second largest eigenvalue µ2.We construct a vector v = (v0, v1, . . . , vm−1) where
v0 = 1
vl =
ui
(n − 1)iM(i − 1) ≤ l ≤ M(i) − 1 i = 1, . . . , j
uj+1
rM(j) ≤ l ≤ M(j) + r − 1
and let v′= (v, v).
46 T. Høholdt, J. Justesen / Discrete Mathematics 325 (2014) 38–46
Wenote that v′ is orthogonal to the all 1-vector. This can be seen in the followingway: B could be obtained as the quotientmatrix of a symmetricmatrix A (with real numbers as entries) partitioned into blocks Aij of sizes 2, 2(n−1), . . . , 2(n−1)j, 2rsuch that the column sumsof Aij are bij and such that in eachblock the rowsums are equal. For A the vector v′ is an eigenvectorcorresponding to the second largest eigenvalue and since n is the largest eigenvalue with eigenvector the all 1- vector theresult follows.
Let Q be the Laplace matrix Q = A − I . We then have
⟨Qv′, v′⟩
⟨v′, v′⟩≥ λq where λq = n − λ2.
The relation is an equality if and only if v′ is an eigenvector. Here ⟨Qv′, v′⟩ =
(va − vb)
2 where the sum is over all edgesa − b in the graph, and it follows from the construction of v′ that only terms of the form (va − va+1)
2 occur in the sum andthat the number of such terms is at most (n − 1)|Aa|. However if we do the same for a matrix for which B is the quotientmatrix (and |Aa| are maximal) the vector v′ is an eigenvector corresponding to n − µ2 and therefore n − µ2 ≥ n − λ2 andhence λ2 ≥ µ2. The smallest value of µ2 is obtained if r = 0 (this corresponds to x = n − 1 and y = n) and in this caseµ2 = 2
√n − 1 cos( π
j+1 ).If we maintain the constraint that |Bj+1| = r , we can vary z (and thus y and x) to get the best second eigenvalue, which
then becomes a lower bound on λ between the valuesM(j) and M(j + 1).Next we consider the bounds in Section 4. For a given graph with adjacency matrix A of size m, the nodes are divided
into an initial subset A0 with 2a nodes and at least ad branches as described in Section 4. We then take subsets A1, A2, . . . atdistance 1, 2, . . . , from the initial subset, and the vector v is now an eigenvector for C ′. From here we proceed as above toget the largest possible graph for the given second eigenvalue. Thus we get an upper bound on m/a or a lower bound on a,the size of the initial subset. It then follows that the minimum distance of the graph code satisfies D ≥ ad.
It follows that if a code satisfies the lower bound with equality, any minimum distance codeword has its support on asubgraph of size a, and the distance classes from any of these subgraphs have sizes a(n − d), a(n − d)(n − 1), . . . .
7. Conclusion
We have given an extended lower bound on the minimum distance of graph codes. The bound depends on the use ofunderlying (often bipartite) graphs with small second eigenvalues. We have provided a tight upper bound on the size ofgraphs with given values of this parameter. The minimum distance can meet the lower bound in many cases. We havedemonstrated that the relevant subgraphs of planes and N-gons can be found as subfield graphs or unions of such graphs.
Acknowledgments
The authors gratefully acknowledge the support of the Danish National Research Foundation and the National ScienceFoundation of China (Grant No. 11061130539) for the Danish–Chinese Center for Applications of Algebraic Geometry inCoding and Cryptography. We also want to thank the anonymous referees for helpful comments and suggestions.
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