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Parameters of two classes of LCD BCH codesChengju Li, Cunsheng Ding, Hao Liu

Abstract

BCH codes are widely employed in data storage, communications systems, and consumer electronics. A newapplication of LCD codes in cryptography was found recently. It is known that LCD codes are asymptoticallygood. LCD cyclic codes were referred to as reversible cycliccodes in the literature. The objective of this paperis to construct a class of LCD codes, which are also BCH codes.The dimensions and the minimum distances ofthese LCD cyclic codes are investigated. We will also study aclass of reversible BCH codes proposed in [13]and extend the results on their parameters. As a byproduct, the parameters of some primitive BCH codes will beanalysed. Some of the codes are optimal or have the best knownparameters.

Index Terms

BCH codes, cyclic codes, LCD codes, reversible cyclic codes.

I. INTRODUCTION

Let GF(q) be a finite field of sizeq. An [n,k,d] linear codeC over GF(q) is a linear subspace ofGF(q)n with dimensionk and minimum (Hamming) distanced. A linear codeC over GF(q) is called anLCD code (linear code with complementary dual)if C ∩C⊥ = {0}, whereC⊥ denotes the (Euclidean)dual of C and is defined by

C⊥ = {(b0,b1, . . . ,bn−1) ∈ GF(q)n :

n−1

∑i=0

bici = 0 for all (c0,c1, . . . ,cn−1) ∈ C}.

An [n,k] linear codeC is calledcyclic if (c0,c1, . . . ,cn−1) ∈ C implies (cn−1,c0,c1, . . . ,cn−2) ∈ C . Byidentifying any vector(c0,c1, . . . ,cn−1) ∈ GF(q)n with

c0+c1x+c2x2+ · · ·+cn−1xn−1 ∈ GF(q)[x]/(xn−1),

A code C of length n over GF(q) corresponds to a subset of GF(q)[x]/(xn− 1). Then C is a cycliccode if and only if the corresponding subset is an ideal of GF(q)[x]/(xn−1). Note that every ideal ofGF(q)[x]/(xn−1) is principal. Then there is a monic polynomialg(x) of the smallest degree such thatC = 〈g(x)〉 and g(x) | (xn − 1). In addition, g(x) is unique and called thegenerator polynomial, andh(x) = (xn−1)/g(x) is referred to as thecheck polynomialof C .

Let f (x) = ftxt + ft−1xt−1+ · · ·+ f1x+ f0 be a polynomial over GF(q) with ft 6= 0 and f0 6= 0. Thereciprocal f (x) of f (x) is defined by

f (x) = f−10 xt f (x−1).

Then we have the following lemma that characterizes LCD cyclic codes over finite fields.

Lemma 1. Let C be a cyclic code overGF(q) with generator polynomial g(x). Then the followingstatements are equivalent.

1) C is an LCD code.

C. Ding’s research was supported by the Hong Kong Research Grants Council, under Grant No. 16301114.C. Li is with the School of Computer Science and Software Engineering, East China Normal University, Shanghai, 200062, China (email:

lichengju1987@163.com).C. Ding is with the Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear

Water Bay, Kowloon, Hong Kong, China (email: cding@ust.hk).H. Liu is with the Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water

Bay, Kowloon, Hong Kong, China (email: hliuar@connect.ust.hk).

2

2) g(x) is self-reciprocal, i.e., g(x) = g(x).3) β−1 is a root of g(x) for every rootβ of g(x).

LCD cyclic codes over finite fields were calledreversible codesand studied by Massey [14]. Masseyshowed that some cyclic LCD codes over finite fields are BCH codes, and made a comparison betweenLCD codes and non-LCD codes [14]. He also demonstrated that asymptotically good LCD codes exist[15]. Yang and Massey gave a necessary and sufficient condition for a cyclic code to have a complementarydual [20]. Using the hull dimension spectra of linear codes,Sendrier showed that LCD codes meet theasymptotic Gilbert-Varshamov bound [18]. Esmaeili and Yari analysed LCD codes that are quasi-cyclic[11]. Muttoo and Lal constructed a reversible code over GF(q) [17]. Tzeng and Hartmann proved that theminimum distance of a class of reversible cyclic codes is greater than the BCH bound [19]. Dougherty,Kim, Ozkaya, Sok and Sole developed a linear programming bound on the largest size of an LCD codeof given length and minimum distance [10]. Carlet and Guilley investigated an application of LCD codesagainst side-channel attacks, and presented several constructions of LCD codes [4]. There are two knownclasses of reversible cyclic codes [16, p. 206], which are Melas’s double-error correcting binary codeswith parameters[2m−1,2m−2m,d ≥ 5] and Zetterberg’s double-error correcting binary codes of length2ℓ+1. A well rounded treatment of reversible cyclic codes was given in [9]. In addition, Boonniyomaand Jitman gave a study on linear codes with Hermitian complementary dual [3].

The objective of this paper is to construct a class of primitive LCD codes, which are also BCH codes, andinvestigate their dimensions and minimum distances. As a byproduct, the parameters of several primitiveBCH codes, some of which are not narrow-sense, will be analysed. In addition, we will also study aclass of reversible cyclic codes proposed in [13] and extendthe results on their parameters. Accordingto the tables of best known linear codes (referred to as theDatabaselater) maintained by Markus Grasslat http://www.codetables.de/ and the tables of best cycliccodes documented in [6], some of the codespresented in this paper are optimal in the sense that they have the best possible parameters.

II. q-CYCLOTOMIC COSETS MODULOn

To deal with cyclic codes of lengthn over GF(q), we have to study the canonical factorization ofxn−1 over GF(q). To this end, we need to introduceq-cyclotomic cosets modulon. Note thatxn−1has no repeated factors over GF(q) if and only if gcd(n,q) = 1. Throughout this paper, we assume thatgcd(n,q) = 1.

Let Zn = {0,1,2, · · · ,n−1} denote the ring of integers modulon. For anys∈ Zn, the q-cyclotomiccoset of s modulo nis defined by

Cs= {s,sq,sq2, · · · ,sqℓs−1} modn⊆ Zn,

whereℓs is the smallest positive integer such thats≡ sqℓs (mod n), and is the size of theq-cyclotomiccoset. The smallest integer inCs is called thecoset leaderof Cs. Let Γ(n,q) be the set of all the cosetleaders. We have thenCs∩Ct = /0 for any two distinct elementss and t in Γ(n,q), and

⋃

s∈Γ(n,q)

Cs= Zn. (1)

Hence, the distinctq-cyclotomic cosets modulon partitionZn.Let m= ordn(q), and letα be a generator of GF(qm)∗. Put β = α(qm−1)/n. Thenβ is a primitiven-th

root of unity in GF(qm). The minimal polynomialms(x) of βs over GF(q) is the monic polynomial of thesmallest degree over GF(q) with βs as a zero. It is now straightforward to prove that this polynomial isgiven by

ms(x) = ∏i∈Cs

(x−βi) ∈ GF(q)[x],

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which is irreducible over GF(q). It then follows from (1) that

xn−1= ∏s∈Γ(n,q)

ms(x)

which is the factorization ofxn−1 into irreducible factors over GF(q). This canonical factorization ofxn−1 over GF(q) is necessary for the study of cyclic codes.

III. A CONSTRUCTION OFLCD CYCLIC CODES OVERGF(q) OF LENGTH n

Throughout this paper, letn be a positive integer with gcd(n,q)=1 andm=ordn(q). Let α be a generator

of GF(qm)∗ and putβ = αqm−1

n . Then β is a primitive n-th root of unity. Letmi(x) denote the minimalpolynomial of βi over GF(q). We usei modn to denote the unique integer in the set{0,1, . . . ,n−1} ,which is congruent toi modulon. We also assume thatmi(x) :=mi modn(x). Let lcm(u,v) denote the leastcommon multiple ofu andv.

For any integerδ with 2≤ δ ≤ n, define

g(q,n,δ,b)(x) = lcm(mb(x),mb+1(x), · · · ,mb+δ−2(x)). (2)

Let C(q,n,δ,b) denote the cyclic code of lengthn with generator polynomialg(q,n,δ,b)(x). ThenC(q,n,δ,b) iscalled a BCH code withdesigned distanceδ. We call C(q,n,δ,b) a narrow-sense BCH codeif b= 1 andnon-narrow-sense BCH codeotherwise. Whenn= qm−1, C(q,n,δ,b) is called aprimitive BCH code. Formore information on BCH codes, we refer the reader to [12] and[16].

Below we present a construction of a class of LCD codes which are also BCH codes. For any integerδ with 2≤ δ ≤ n, define

g(x) =

lcm(

x+1,g(q,n,δ, n2+1)(x),g(q,n,δ, n

2−(δ−1))(x)

), if n is even;

lcm(

g(q,n,δ, n+12 )(x),g

(q,n,δ, n+1

2 −(δ−1))(x)

), if n is odd.

(3)

It can be verified that

g(x) =

g(

q,n,2δ, n2−(δ−1)

)(x), if n is even;

g(q,n,2δ−1, n+1

2 −(δ−1))(x), if n is odd.

(4)

Let C be a cyclic code of lengthn with the generator polynomialg(x). In this paper, we always assumethat 2δ−1< n if n is even and 2δ−2< n if n is odd, which can ensure thatg(x) 6= xn−1 andC 6= {0}.It is easy to check thatg(x) is self-reciprocal in both cases. It then follows from Lemma1 that C is anLCD code of lengthn, which is also a BCH code.

Although BCH codes are not good asymptotically, they are among the best linear codes when the lengthof the codes is not very large [6, Appendix A]. So far, we have very limited knowledge of BCH codes, asthe dimension and minimum distance of BCH codes are in general open, in spite of some recent progress[7], [8]. As pointed out by Charpin in [5], it is a well-known hard problem to determine the minimumdistance of narrow-sense BCH codes. However, in some special cases the minimum distance is known.

The following result is sometimes useful in determining theminimum distance of the narrow-senseBCH codes [2, p. 247].

Lemma 2. For a narrow-sense BCH codeC(q,n,δ,1) over GF(q) of length n with designed distanceδ, itsminimum distance d= δ if δ divides n.

The following corollary is a generalization of Lemma 2 and will be employed later.

Corollary 3. Let C(q,n,δ,b) be the BCH code over GF(q) of length n with designed distanceδ. Then itsminimum distance d= δ if δ dividesgcd(n,b−1).

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Proof: Denote

c(x) =xn−1

xn/δ −1= x(δ−1) n

δ + · · ·+xnδ +1.

Sinceδ | (b−1), we haveδ ∤ j and

c(β j) = 0 for j = b,b+1, . . . ,b+δ−2,

whereβ is a primitiven-th root of unity. It then follows thatc(x) ∈ C(q,n,δ,b). It is clear that the Hammingweight of c(x) is equal toδ. Thend ≤ δ and this completes the proof.

IV. PARAMETERS OF THE PRIMITIVE NARROW-SENSEBCH CODESC(q,n,δ,1)

In this section, we always assume thatn = qm− 1 and u is an integer with 1≤ u ≤ q− 1. Denotem= ⌈m

2 ⌉ and

T :=

{qm, if m is odd;

2qm, if m is even.(5)

Lemma 4. [1], [21] For any i with 1≤ i ≤ T and q∤ i, i is a coset leader of the cyclotomic coset Ci and|Ci|= m for all i in the range1≤ i ≤ T except i= q

m2 +1 with |C

qm2 +1

|= m2 when m is even.

We present the cardinality of each cyclotomic cosetCj and characterize all coset leadersj in the range1≤ j ≤ uqm in the following proposition, wherem≥ 5 is an odd integer.

Proposition 5. Let m≥ 3 be an odd integer and let j be an integer with1≤ j ≤ uqm and q∤ j, where1≤ u≤ q−1 and m= m+1

2 . Then the following hold.

1) |Cj |= m except j= v(q2+q+1) with |Cv(q2+q+1)|= 1 when m= 3, where1≤ v≤ q−2.2) j is a coset leader of the cyclotomic coset Cj except j∈ J1∪J2, where

J1 = { jmqm+ j1q+ j0 : 1≤ jm≤ u−1,0≤ j1 < jm,1≤ j0 ≤ q−1} (6)

andJ2 = { jmqm+ jm−1qm−1+ j0 : 1≤ jm ≤ u−1,1≤ jm−1 ≤ q−1,1≤ j0 ≤ jm}. (7)

3) |J1∪J2|= (u2−u)(q−1).

Proof: For any j with 1≤ j ≤ uqm, let ℓ = |Cj |. Sincem is odd, we have 1≤ ℓ ≤ m3 if ℓ < m. For

m≥ 9, one can check thatj < jqℓ < n for all 1≤ j ≤ uqm,

which means thatjqℓ ≡ j (mod n)

doesn’t hold for anyℓ < m. Thus we have|Cj |= m if m≥ 9.For m= 3,5 or 7, if |Cj |< m, then |Cj |= 1, which is equivalent to saying that

α j ∈ GF(q)∗, i.e., j ≡ 0 (modqm−1q−1

),

whereα is a generator of GF(qm)∗. Whenm= 5 or 7, this is a contradiction because 0< j ≤ uqm< qm−1q−1 .

The desired conclusion on the cardinality of the cyclotomiccosetCj then follows.Below we characterize all coset leadersj in the range 1≤ j ≤ uqm. To this end, we have to find all

integers j satisfying j ∈Ci , i.e.,jqℓ modn= i (8)

5

for some integerℓ with 1≤ ℓ≤ m−1 and some integeri < j. Let i and j be two integers withq ∤ i,q ∤ j,and i < j ≤ uqm. By Lemma 4, j is a coset leader if 1≤ j ≤ qm andq ∤ j, so we can further assume thatj ≥ qm+1. Then we have the twoq-adic expansions

i = imqm+ im−1qm−1+ · · ·+ i1q+ i0

andj = jmqm+ jm−1qm−1+ · · ·+ j1q+ j0,

where 1≤ i0, j0 ≤ q−1, 1≤ jm ≤ u−1, and 0≤ im ≤ jm.Case 1:When 1≤ ℓ≤ m−2, it is easy to check thati < jqℓ < n, so (8) doesn’t hold.Case 2:Whenℓ= m−1, we have

jqℓ = jmqm+ jm−1qm−1+ · · ·+ j1qm+ j0qm−1

by noting thatm= m+12 andm= 2m−1. Then

jqℓ modn= jm−1qm−1+ · · ·+ j2qm+1+ j1qm+ j0qm−1+ jm.

By (8), we obtainjm = i0, jm−1 = jm−2 = · · ·= j2 = 0, j1 = im, j0 = im−1. (9)

Thus j = jmqm+ j1q+ j0.Notice thati < j. Then im ≤ jm. We assert that the equality doesn’t hold. Otherwise, it follows from

(9) andi < j that j0 = im−1 = 0, which is a contradiction. We then deduce that 0≤ j1 = im< jm≤ u−1.Denote

J1 = { jmqm+ j1q+ j0 : 1≤ jm ≤ u−1,0≤ j1 ≤ jm−1,1≤ j0 ≤ q−1}.

Then (8) holds if and only ifj ∈ J1 for ℓ= m−1.Case 3:Whenℓ= m, we have

jqℓ = jmqm+1+ jm−1qm+ · · ·+ j1qm+1+ j0qm.

Thenjqℓ modn= jm−2qm−1+ · · ·+ j1qm+1+ j0qm+ jmq+ jm−1.

By (8), we obtainjm = i1, jm−1 = i0, jm−2 = · · ·= j2 = j1 = 0, j0 = im. (10)

Thus j = jmqm+ jm−1qm−1+ j0.Case 3.1:If im < jm, it then follows from (10) that (8) holds if and only ifj ∈ J21, where

J21 = { jmqm+ jm−1qm−1+ j0 : 1≤ jm ≤ u−1,1≤ jm−1 ≤ q−1,1≤ j0 < jm}.

Case 3.2:If im = jm and im−1 < jm−1, it then follows from (10) that (8) holds if and only ifj ∈ J22,where

J22 = { jmqm+ jm−1qm−1+ j0 : 1≤ jm ≤ u−1,1≤ jm−1 ≤ q−1, j0 = jm ≥ 1}.

Case 3.3:If im = jm, im−1 = jm−1, im−2 = · · ·= i1 = 0, andi0 < j0, it then follows from (10) that (8)holds if and only if j ∈ J23, where

J23= { jmqm+ jm−1qm−1+ j0 : 1≤ jm≤ u−1,1≤ jm−1 ≤ q−1,1≤ i0 < j0 = jm}.

Denote

J2 = J21∪J22∪J23 = { jmqm+ jm−1qm−1+ j0 : 1≤ jm ≤ u−1,1≤ jm−1 ≤ q−1,1≤ j0 ≤ jm}.

Then (8) holds if and only ifj ∈ J2 for ℓ= m.

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Case 4:Whenm+1≤ ℓ≤ m−1, let ℓ= m+ ε, where 1≤ ε ≤ m−2. Then we have

jqℓ = jmq2m+ε + · · ·+ jm−ε−1qm+ jm−ε−2qm−1+ · · ·+ j0qm+ε.

Thenjqℓ modn= jm−ε−2qm−1+ · · ·+ j0qm+ε + jmqε+1+ · · ·+ jm−ε−1.

Note that j0 ≥ 1. Then jqℓ modn> i, which implies that (8) is impossible in this case.Collecting all the discussions in Cases 1, 2, 3, and 4, we get the desired conclusion on coset leaders.

Note that|J1|= |J2|=u(u−1)

2 (q−1). It is clear thatJ1∩J2 = /0, so|J1∪J2|= (u2−u)(q−1). This completesthe proof.

For convenience, below we assume thatm≥ 5 whenm is odd, while the case thatm= 3 can be similarlydealt with.

Theorem 6. Let m≥ 5 be an odd integer andδ = uqm+1

2 +1, where1≤ u≤ q−1. Then the codeC(q,n,δ,1)has length n= qm−1, dimension

k= qm−1− (uqm−1

2 −u2+u)(q−1)m,

and minimum distance d≥ δ. Furthermore, the generator polynomial is given by

g(q,n,δ,1)(x) = ∏1≤ j≤uq

m+12

q∤ j , j 6∈J1∪J2

mj(x).

Proof: The desired conclusions follow from Proposition 5 and the BCH bound immediately.

Example 1. 1) When(q,m,u) = (2,5,1), the codeC(q,n,δ,1) has parameters[31,11,11], which is anoptimal code according to the Database.

2) When(q,m,u) = (2,7,1), the codeC(q,n,δ,1) has parameters[127,71,19], which are the best param-eters for linear codes according to the Database.

The following proposition gives the cardinality of each cyclotomic cosetCj and characterizes all cosetleadersj in the range 1≤ j ≤ uqm, wherem≥ 2 is an even integer.

Proposition 7. Let m≥ 2 be an even integer and let j be an integer with1≤ j ≤ uqm and q∤ j, where1≤ u≤ q−1 and m= m

2 . Then the following hold.1) |Cj |= m except j= v(qm+1) with |Cv(qm+1)|=

m2 , where v= 1,2, . . . ,u−1.

2) j is a coset leader of the cyclotomic coset Cj except j∈ J, where

J = { jmqm+ j0 : j0 < jm≤ u−1,1≤ j0 ≤ u−1}. (11)

3) |J|= (u−1)(u−2)2 .

Proof: Let i and j be two integers withq ∤ i,q ∤ j, and i < j ≤ uqm. Suppose thatj ∈Ci. Then thereexists some integerℓ with 1≤ ℓ≤ m−1 such that

jqℓ modn= i. (12)

By Lemma 4, j is a coset leader if 1≤ j ≤ 2qm and q ∤ j, so we can further assume thatj ≥ 2qm+1.Then we have the twoq-adic expansions

i = imqm+ im−1qm−1+ · · ·+ i1q+ i0

andj = jmqm+ jm−1qm−1+ · · ·+ j1q+ j0,

where 1≤ i0, j0 ≤ q−1, 2≤ jm ≤ u−1, and 0≤ im ≤ jm.

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Case 1:When 1≤ ℓ≤ m−1, it is easy to check thati < jqℓ < n, so (12) doesn’t hold.Case 2:Whenℓ= m, we have

jqℓ = jmqm+ jm−1qm−1+ · · ·+ j1qm+1+ j0qm.

Thenjqℓ modn= jm−1qm−1+ · · ·+ j1qm+1+ j0qm+ jm.

By (8), we obtainjm = i0, jm−1 = jm−2 = · · ·= j1 = 0, j0 = im. (13)

Thus j = jmqm+ j0.Case 2.1:If im < jm, it then follows from (13) that (12) holds if and only ifj ∈ J, where

J = { jmqm+ j0 : im< jm,2≤ jm ≤ u−1,1≤ j0 = im < u−1}.

Case 2.2:If im = jm, we deduce thatim−1 = · · ·= i1 = 0 and i0 < j0 from i < j. Then

i0 < j0 = im = jm = i0,

which is a contradiction. Thus (12) doesn’t hold.Case 3:Whenm+1≤ ℓ≤ m−1, let ℓ= m+ ε, where 1≤ ε ≤ m−1. Then

jqℓ = jmq2m+ε + · · ·+ jm−εqm+ jm−ε−1qm−1+ · · ·+ j0qm+ε.

Thenjqℓ modn= jm−ε−1qm−1+ · · ·+ j1qm+ε+1+ j0qm+ε + jmqε + jm−1qε−1+ · · ·+ jm−ε.

Note that j0 ≥ 1. Then jqℓ modn> i, which implies that (12) is impossible in this case.Summarizing all the discussions in Cases 1, 2, and 3, we get the desired conclusion of 2). It is easy to

see thatjqℓ modn> j

in both Cases 1 and 3. Then we have|Cj |= m if |Cj |< m. Moreover, it follows from Case 2 that

j = jmqm+ j0 = j0qm+ jm and j0 = jm.

Then we proved 1). It is clear thatJ =(u−1)(u−1)

2 . This completes the proof.

Theorem 8. Let m≥ 2 be an even integer andδ = uqm2 +1. Then the codeC(q,n,δ,1) has length n= qm−1,

dimension

k= qm−1−uqm2 −1(q−1)m+

(u−1)2

2m,

and minimum distance d≥ δ. When u= 1, we have d= δ. Furthermore, the generator polynomial is givenby

g(q,n,δ,1)(x) = ∏1≤ j≤uq

m2

q∤ j , j 6∈J

mj(x).

Proof: When u = 1, it is clear thatδ|n. The desired conclusions then follow from Lemma 2,Proposition 7 and the BCH bound.

Example 2. 1) When (q,m,u) = (2,4,1), the codeC(q,n,δ,1) has parameters[15,7,5], which is anoptimal code according to the Database.

2) Let (q,m) = (3,4). When u∈ {1,2}, the codeC(q,n,δ,1) has parameters[80,56,10] and [80,34,20],respectively. The former has the best parameters for linearcodes according to the Database.

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V. PARAMETERS OF THELCD CODE C WHEN q IS ODD

In this section, we always assume thatq is odd,u is an integer with 1≤ u ≤ q−1, andn= qm−1.The following proposition will be useful.

Proposition 9. Let m≥ 2, n= qm−1, and n= n2, where q is odd. Then we have the following.

1) |Cn+i |= |Ci|= |C−i |= |Cn−i |.2) |Cn+qi|= |Cn+i | and |Cn−qi|= |Cn−i |.3) Ci =Cj if and only if Cn+i =Cn+ j .4) Ci =Cj if and only if Cn−i =Cn− j .

Proof: Note thatq is odd. It is clear thatn2± i ≡ (

n2± j)qℓ (mod n),

is equivalent toi ≡ jqℓ (mod n)

for any ℓ with 0≤ ℓ≤ m−1. The desired conclusions then follow.Let 1≤ u≤ q−1 be an integer. Define

J+(q,n,u) =⋃

1≤ j≤uqm

Cn+ j andJ−(q,n,u) =⋃

1≤ j≤uqm

Cn− j ,

whereq is odd and ¯n= n2. It can be deduced from Proposition 9 thatCn+i 6=Cn+ j andCn−i 6=Cn− j if and

only if Ci 6=Cj . The following corollary then follows from Propositions 5 and 7 directly.

Corollary 10. Let j be an integer with1≤ j ≤ uqm, q odd, and n= qm−1.1) If m≥ 5 is odd, then|Cn+ j |= |Cn− j |= m and

|J+(q,n,u)|= |J−(q,n,u)|= (uqm−1−u2+u)(q−1)m.

2) If m≥ 2 is even, then|Cn+ j |= |Cn− j |=m except j= v(qm+1) with |Cn+v(qm+1)|= |Cn−v(qm+1)|=m2 ,

where v= 1,2, . . . ,u−1. In this case,

|J+(q,n,u)|= |J−(q,n,u)|= uqm−1(q−1)m−(u−1)2

2m.

Theorem 11. Let m≥ 2 be an integer andδ = uqm+1.1) If m≥ 5 is odd, thenC(q,n,δ, n

2+1) and C(q,n,δ, n2−(δ−1)) both have length n= qm−1, dimension

k= qm−1− (uqm−1

2 −u2+u)(q−1)m,

and minimum distance d≥ δ. In addition, the generator polynomials are given by

g(q,n,δ, n2+1)(x) = ∏

1≤ j≤uqm+1

2

q∤ j , j 6∈J1∪J2

mn2+ j(x) and g(q,n,δ, n

2−(δ−1))(x) = ∏1≤ j≤uq

m+12

q∤ j , j 6∈J1∪J2

mn2− j(x).

2) If m≥ 2 is even, thenC(q,n,δ, n2+1) and C(q,n,δ, n

2−(δ−1)) both have length n= qm−1, dimension

qm−1−uqm2 −1(q−1)m+

(u−1)2

2m,

and minimum distance d≥ δ. In addition, the generator polynomials are given by

g(q,n,δ, n2+1)(x) = ∏

1≤ j≤uqm2

q∤ j , j 6∈J

mn2+ j(x) and g(q,n,δ, n

2−(δ−1))(x) = ∏1≤ j≤uq

m2

q∤ j , j 6∈J

mn2− j(x).

9

Proof: The proof can be presented using Corollary 10 and the BCH bound directly, and is omittedhere.

Example 3. 1) Let (q,m)= (3,5). When u∈ {1,2}, the codeC(q,n,δ,1) has parameters[242,152,d≥ 28]and [242,82,d ≥ 55], respectively.

2) Let (q,m)= (4,4). When u∈{1,2,3}, the codeC(q,n,δ,1) has parameters[255,207,d≥17], [242,161,d≥33], and [242,119,d ≥ 49], respectively.

The following theorem gives the dimension and and the lower bound on the minimum distance of thecodeC(q,n,δ, n

2+1) (or C(q,n,δ, n2−(δ−1))), whereδ = qt −1 for an integert with 1≤ t ≤ m.

Theorem 12. Let m≥ 2 be an integer andδ = qt −1, where q is odd and t is an integer with1≤ t ≤ m.ThenC(q,n,δ, n

2+1) and C(q,n,δ, n2−(δ−1)) both have length n= qm−1, dimension

k= qm−1− (qt −qt−1−1)m,

and minimum distance d≥ qt −1. In addition, the generator polynomials are given by

g(q,n,δ, n2+1)(x) = ∏

1≤ j≤qt−2q∤ j

mn2+ j(x) and g(q,n,δ, n

2−(δ−1))(x) = ∏1≤ j≤qt−2

q∤ j

mn2− j(x).

Proof: It is known thatCn+i 6=Cn+ j andCn−i 6=Cn− j if and only if Ci 6=Cj . One can easily get thegenerator polynomialsg(q,n,δ, n

2+1)(x) and g(q,n,δ, n2−(δ−1))(x) from Lemma 4, and their degrees are equal

to (qt −2− qt−qq )m. Then the desired conclusion follows from the BCH bound.

A. Parameters ofC when m is oddThe following proposition plays an important role in determining the dimension of the BCH codeC

whenm≥ 5 is odd andδ = uqm+1

2 +1, where 1≤ u≤ q−1.

Proposition 13. For odd m≥ 5, we have

J+(q,n,u)∩J−(q,n,u) =⋃

j∈JO

Cn+ j ∪Cn− j ,

where the union is disjoint and

JO= { jmqm+ jm−1qm−1+(q−1)(qm−2+· · ·+q2+q)+ j0 : 0≤ jm≤u−1,0≤ jm−1≤q−2,q−u≤ j0≤q−1}.

Moreover,|J+(q,n,u)∩J−(q,n,u)|= 2u2(q−1)m.

Proof: Suppose thata∈ J+(q,n,u)∩J−(q,n,u). Then there exist 1≤ i ≤ uqm and 1≤ j ≤ uqm such that

a∈Cn2+i =Cn

2− j ,

which implies thatn2+ i ≡ (

n2− j)qℓ (mod n) and i + jqℓ ≡ 0 (mod n) (14)

for some 1≤ ℓ≤ m−1.By Proposition 9, we can further assume thatq ∤ i andq ∤ j. Then we have theq-adic expansions

i = imqm+ im−1qm−1+ · · ·+ i1q+ i0

andj = jmqm+ jm−1qm−1+ · · ·+ j1q+ j0,

where 0≤ im, jm≤ u−1, 1≤ i0, j0 ≤ q−1, and 0≤ ik, jk ≤ q−1 for all k with 1≤ k≤ m−1.

10

Case 1:When 1≤ ℓ≤ m−2, it is easy to check that 0< i+ jqℓ < n by noticing thatjm≤ u−1< q−1,so i + jqℓ ≡ 0 (mod n) doesn’t hold.

Case 2:Whenℓ= m−1, it can be verified thati + jqℓ ≡ ∆ (mod n), where

∆ = jm−1qm−1+ · · ·+ j2qm+1+( j1+ im)qm+( j0+ im−1)q

m−1+ im−2qm−1+ · · ·+ i1q+(i0+ jm).

It is clear that 0< ∆ < 2n. It then follows from (14) that∆ = n. Thus

jm−1 = · · ·= j2 = j1+ im = j0+ im−1 = im−2 = · · ·= i1 = i0+ jm = q−1.

Thenj = jmqm+(q−1)(qm−1+qm−2+ · · ·+q2)+ j1q+ j0,

where0≤ jm ≤ u−1, q−u≤ j1 ≤ q−1, and 1≤ j0 ≤ q−1.

Case 3:Whenℓ= m, we havei + jqℓ ≡ ∆ (mod n), where

∆ = jm−2qm−1+ · · ·+ j1qm+1+( j0+ im)qm+ im−1qm−1+ · · ·+ i2q2+( jm+ i1)q+( jm−1+ i0).

Notice that 0< ∆ < 2n. It then follows from (14) that∆ = n. Thus

jm−2 = · · ·= j1 = j0+ im = im−1 = · · ·= i2 = jm+ i1 = jm−1+ i0 = q−1.

Thenj = jmqm+ jm−1qm−1+(q−1)(qm−2+ · · ·+q2+q)+ j0,

where0≤ jm ≤ u−1, 0≤ jm−1 ≤ q−2, andq−u≤ j0 ≤ q−1.

Case 4:When m+1≤ ℓ≤ m−1, denoteℓ= m+ ε, where 1≤ ε ≤ m−2. Theni + jqℓ ≡ ∆ (mod n),where

∆ = jm−ε−2qm−1+ · · ·+ j0qm+ε + imqm+ im−1qm−1+ · · ·+ iε+2qε+2

+ (iε+1+ jm)qε+1+ · · ·+(i1+ jm−ε)q+(i0+ jm−ε−1).

It is easy to see that the coefficient ofqm in the q-adic expansion of∆ is less thanq−1. Thus we have0< ∆ < n, which means that (14) is impossible.

Summarizing the discussions in Cases 1, 2, 3, and 4, we get thedesired conclusions.The following result gives the dimension of the LCD codeC when m≥ 5 is odd andδ = uq

m+12 +1,

where 1≤ u≤ q−1.

Theorem 14. Let m≥ 5 be an odd integer, q odd, andδ = uqm+1

2 +1, where1≤ u≤ q−1. Let C be theprimitive BCH code with generator polynomial g(x) given in(3). ThenC is an LCD cyclic code and haslength n= qm−1, dimension

k= qm−2−2(uqm−1

2 −2u2+u)(q−1)m,

and minimum distance d≥ 2δ. In addition, the generator polynomial is given by

g(x) = (x−1) ∏1≤ j≤uq

m+12

q∤ j , j 6∈J1∪J2∪JO

mn2+ j(x)mn

2− j(x). (15)

Proof: By Proposition 13 and Theorem 11, it is easy to see that the generator polynomialg(q,n,δ) is

given by (15), and its degree is equal to 1+2(uqm−1

2 −2u2+u)(q−1)m. The dimensionk = qm−1−deg(g(x)) then follows. Notice that the polynomialg(x) has the rootsβi for all i in the set

{n2− (δ−1), . . . ,

n2−1,

n2,n2+1, . . . ,

n2+(δ−1)},

11

whereδ = uqm+1

2 +1. Then we deduce thatd ≥ 2δ by the BCH bound.

Example 4. Let (q,m) = (3,7). When u∈ {1,2}, the codeC has parameters[2186,1457,d ≥ 164], and[2186,841,d ≥ 326], respectively.

B. Parameters ofC when m is evenTo investigate the parameters of the LCD cyclic codeC whenm≥ 2 is even, we will need the following

conclusion.

Proposition 15. For even m≥ 2, we have

J+(q,n,u)∩J−(q,n,u) =⋃

j∈JE

Cn− j ,

where the union is disjoint and

JE = { jmqm+(q−1)(qm−1+qm−2+ · · ·+q)+ j0 : 0≤ jm≤ u−1 and q−u≤ j0 ≤ q−1}.

Moreover,|J+(q,n,u)∩J−(q,n,u)|= u2m.

Proof: For a∈ J+(q,n,θ)∩J−

(q,n,θ), there exist 1≤ i ≤ θ and 1≤ j ≤ θ such that

a∈Cn2+i =Cn

2− j .

With the same argument in Proposition 13, we have

i + jqℓ ≡ 0 (mod n)

for some 1≤ ℓ≤ m−1.By Proposition 9, we also assume thatq ∤ i andq ∤ j. For i, j ≤ uqm, let

i = imqm+ im−1qm−1+ · · ·+ i1q+ i0

andj = jmqm+ jm−1qm−1+ · · ·+ j1q+ j0,

where 0≤ im, jm≤ u−1, 1≤ i0, j0 ≤ q−1, and 0≤ ik, jk ≤ q−1 for all 1≤ k≤ m−1.Case 1:When 1≤ ℓ≤ m−1, we easily see that 0< i + jqℓ < n as jm ≤ u−1< q−1, which implies

that i + jqℓ ≡ 0 (mod n) does not hold.Case 2:Whenℓ= m, it can be verified that

i + jqℓ ≡ ∆ (mod n),

where∆ = jm−1qm−1+ · · ·+ j1qm+1+( j0+ im)q

m+ im−1qm−1+ · · ·+ i1q+(i0+ jm).

Notice that 0< ∆ < 2n. If ∆ ≡ 0 (mod n), then∆ = n and

jm−1 = · · ·= j1 = j0+ im = im−1 = · · ·= i1 = i0+ jm = q−1.

Thusj = jmqm+(q−1)(qm−1+qm−2+ · · ·+q)+ j0,

where0≤ jm ≤ u−1 andq−u≤ j0 ≤ q−1.

Case 3:Whenm+1≤ ℓ≤m−1, letℓ= m+ε, where 1≤ ε≤ m−1. Then one can check thati+ jqℓ ≡∆(mod n), where

∆ = jm−ε−1qm−1+ · · ·+ j0qm+ε + imqm+ · · ·+ iε+1qε+1+(iε + jm)qε + · · ·+(i0+ jm−ε).

12

Note that the coefficient ofqm in theq-adic expansion of∆ is equal toim≤ u−1< q−1. Then 0< ∆ < n,which means that

(i + jqℓ) modn= ∆ 6≡ 0 (mod n).

The desired conclusion then follows from the discussions inCases 1, 2, and 3.

Theorem 16. Let q be odd andδ = uqm2 +1, where1 ≤ u ≤ q−1. Let m= 2 and 1 ≤ u ≤ q−1

2 or letm≥ 4 be an even integer. LetC be the primitive BCH code with generator polynomial g(x) given by(3).Then the LCD codeC has length n= qm−1, dimension

k= qm−2−2uqm2 −1(q−1)m+(u−1)2m+u2m,

and minimum distance d≥ 2δ. In addition, the generator polynomial is given by

g(x) = ∏1≤ j≤uq

m2

q∤ j , j 6∈J

mn2+ j(x) ∏

1≤ j≤uqm2

q∤ j , j 6∈J∪JE

mn2− j(x).

Proof: With the help of Proposition 15 and Theorem 11, the proof is very similar to that of Theorem14 and is omitted here.

Example 5. When(q,m,u)= (5,2,1), the codeC has parameters[24,9,12], which are the best parametersfor linear codes according to the Database.

Corollary 17. Let u= 1 andδ= qm2 +1, where q≡ 3 (mod 4) and m≡ 2 (mod 4). Then the true minimum

distance of the codeC presented in Theorem 16 is equal to2δ.

Proof: Note thatb= n2 − δ+1. It is easy to check that 2δ | gcd(n,b−1) in this case. The desired

result then follows from Corollary 3.

Example 6. When(q,m,u) = (7,2,1), the codeC has parameters[48,25,16], which are the best param-eters for linear codes according to the Database.

C. Parameters ofC with designed distance4 or qt −1, where1≤ t ≤ mWhen q is odd, the true minimum distances of the LCD codesC can be determined in some special

cases. The following theorem presents the parameters of thecodeC when the designed distance is equalto 2δ = 4.

Theorem 18. Let q≥ 5 be odd and m≥ 2. ThenC(q,n,4, n2−1) is an LCD cyclic code and has parameters

[qm−1,qm−2−2m,4] and generator polynomial(x+1)mn2+1(x)mn

2−1(x).

Proof: It follows from Propositions 13 and 15 thatCn2−1 6=Cn

2+1. Note that|Cn2−1|= |Cn

2+1|= |C1|=m.We then deduce that the dimension ofC is qm−2−2m. It is clear thatδ = 2 andd ≥ 2δ = 4 by the BCHbound. We also haved ≤ 4 by the sphere-packing bound. This completes the proof.

The dimension of the LCD codeC is described in the following theorem whenC has designed distance2δ = qt −1 for an integert with 1≤ t ≤ m.

Theorem 19. Let q be odd and m≥ 2. Let C be the primitive BCH code with generator polynomial g(x)given in(3) with designed distance2δ = qt −1, where1≤ t ≤ m. ThenC is an LCD cyclic code and haslength n= qm−1, dimension

k= qm−2− (qt −qt−1−2)m

and minimum distance d≥ qt −1.

Proof: Note thatδ = qt−12 . It then follows from the proofs of Propositions 13 and 15 that

( ⋃

1≤ j≤δ−1

Cn+ j

)⋂( ⋃

1≤ j≤δ−1

Cn− j

)= /0 (16)

13

for any integerm with m≥ 2 andm 6= 3. It can be checked that (16) also holds form= 3.The generator polynomial of the codeC is g(q,n,2δ, n

2−(δ−1))(x). For oddq, it is known thatCn+i 6=Cn+ jandCn−i 6=Cn− j if and only if Ci 6=Cj . By Lemma 4, we have

deg(g(q,n,2δ, n2−(δ−1))(x)) = 1+(qt −qt−1−2)m for 1≤ t ≤ m.

Thus the dimension is obtained. Moreover, the desired result on the minimum distance ofC follows fromthe BCH bound.

Example 7. 1) Let (q,m)= (3,5). When t∈{1,2,3}, the codeC has parameters[242,241,2], [242,221,8]and [242,161,26], and all of them are the best known parameters for linear codes according to theDatabase.

2) Let (q,m) = (3,4). When t∈ {1,2}, the codeC has parameters[80,79,2] and [80,63,8], and bothof them are the best known parameters for linear codes according to the Database.

The following conjecture is supported by experimental dataand Theorem 5 in [16, p. 260].

Conjecture 1. The LCD cyclic codeC presented in Theorem 19 has true minimum distance qt −1.

VI. PARAMETERS OF THELCD CODE C WHEN q IS EVEN

In this section, we always assume thatq is even,u is an integer with 1≤ u≤ q−1, andn= qm−1.

Proposition 20. Let m≥ 2, n= qm−1, and n= n+12 , where q is even. Then we have the following.

1) |Cn+i |= |C2i+1| and |Cn−i |= |C2i−1|.2) C2i+1 =C2 j+1 if and only if Cn+i =Cn+ j .3) C2i−1 =C2 j−1 if and only if Cn−i =Cn− j .

Proof: Sinceq is even and gcd(2,n) = 1, it is clear that

n+12

± i ≡ (n+1

2± j)qℓ (mod n),

which is equivalent to2i ±1≡ (2 j ±1)qℓ (mod n)

for any ℓ with 0≤ ℓ≤ m−1. The desired conclusions then follow.Let 1≤ u≤ q−1 be an integer. Define

J+(q,n,u) =⋃

0≤ j≤uqm/2−1

Cn+ j and J−(q,n,u) =⋃

1≤ j≤uqm/2

Cn− j ,

whereq is even and ¯n= n+12 .

A. Parameters ofC when m is oddIt can be deduced from Proposition 20 thatCn+i 6=Cn+ j if and only if C2i+1 6=C2 j+1 (resp.Cn−i 6=Cn− j

if and only if C2i−1 6=C2 j−1). Let J1 and J2 be the sets of integers that are not coset leaders, which aregiven by (6) and (7). Note that 1≤ 2 j +1≤ uqm−1 if 0 ≤ j ≤ uqm/2−1 and 1≤ 2 j −1≤ uqm−1 if1≤ j ≤ uqm/2. It then follows that

|J+(q,n,u)|= |J−(q,n,u)|= (λ1+λ2)m, (17)

whereλ1 = |{1≤ j ≤ uqm−1 : j is an odd coset leader}| (18)

andλ2 = |{1≤ j ≤ uqm−1 : j ∈ J1∪J2 is odd, the coset leader ofC j is even}|. (19)

Lemma 21. Let q be even, m odd, and n= qm−1. Then the following hold.

14

1)

λ1 =

{uqm/2−q(u2−u)/4−u2(q−1)/4, if u is even;

uqm/2−q(u2−u)/4− (u2−1)(q−1)/4, if u is odd.

2)

λ2 =

{(qu2−qu−u2)/4, if u is even;

(u2−1)(q−1)/4, if u is odd.

Proof: Notice thatq is even. It then follows from Proposition 5 and (18) that

λ1 = uqm/2−|{ j ∈ J1 : j is odd}|− |{ j ∈ J2 : j is odd}|.

By (6) and (7), it is easy to see that

|{ j ∈ J1 : j is odd}| = |{ jmqm+ j1q+ j0 : 1≤ jm ≤ u−1,0≤ j1 < jm,1≤ odd j0 ≤ q−1}|

= q(u2−u)/4

and

|{ j ∈ J2 : j is odd}|=

{u2(q−1)/4, if u is even;

(u2−1)(q−1)/4, if u is odd.

Then we prove the conclusion onλ1.Denote

CL1 = {the coset leader ofC j : j ∈ J1} and CL2 = {the coset leader ofC j : j ∈ J2}.

By the proof of Proposition 5, we have

CL1 = { j1qm+ j0qm−1+ jm : 1≤ jm ≤ u−1,0≤ j1 < jm,1≤ j0 ≤ q−1}

andCL2 = { j0qm+ jmq+ jm−1 : 1≤ jm ≤ u−1,1≤ jm−1 ≤ q−1,1≤ j0 ≤ jm}.

It then follows from (19) that

λ2 = |{ j ∈ J1 : j0 is odd andjm is even}|+ |{ j ∈ J2 : j0 is odd andjm−1 is even}|.

One can easily check that

λ2 =

{(qu2−qu−u2)/4, if u is even;(u2−1)(q−1)/4, if u is odd.

This completes the proof.The following proposition follows from Lemma 21 and (17) directly.

Proposition 22. Let m≥ 5 be odd. Then

|J+(q,n,u)|= |J−(q,n,u)|=

{(uqm/2−qu2/4)m, if u is even;(

uqm/2−q(u2−u)/4)

m, if u is odd.

Theorem 23. Let m≥ 5 be an odd integer, q even, andδ = uqm+1

2 /2+1.1) If u is even, thenC(q,n,δ, n+1

2 ) and C(q,n,δ, n+12 −(δ−1)) both have length n= qm−1, dimension

qm−1− (uqm+1

2 /2−qu2/4)m,

and minimum distance d≥ δ.

15

2) If u is odd, thenC(q,n,δ, n+12 ) and C(q,n,δ, n+1

2 −(δ−1)) both have length n= qm−1, dimension

k= qm−1−(

uqm+1

2 /2−q(u2−u)/4)

m,

and minimum distance d≥ δ.

Proof: The desired conclusions follow from Proposition 22 and the BCH bound directly.

Example 8. 1) When(q,m,u) = (2,7,1), the codeC(q,n,δ, n+12 ) ( or C(q,n,δ, n+1

2 −(δ−1))) has parameters[127,71,19], which are the best parameters for linear codes according tothe Database.

2) Let (q,m) = (4,5). When u∈ {1,2,3}, the codeC(q,n,δ, n+12 ) ( or C(q,n,δ, n+1

2 −(δ−1))) has parameters[1023,863,d ≥ 33], [1023,723,d≥ 65], and [1023,573,d ≥ 97], respectively.

The following conclusion will be employed to determine the dimension of the codeC whenm≥ 5 isodd.

Proposition 24. For odd m≥ 5, we have

J+(q,n,u)∩ J−(q,n,u) =⋃

j∈JO

Cn+( j−1)/2∪Cn−( j+1)/2,

where the union is disjoint and

JO = { jmqm+ jm−1qm−1+(q−1)(qm−2+ · · ·+q2+q)+ j0 : 1≤ jm ≤ u−1,

0≤ even jm−1 ≤ q−2,q−u≤ odd j0 ≤ q−1}.

Moreover,

|J+(q,n,u)∩ J−(q,n,u)|=

{q2u2m, if u is even;q2u(u+1)m, if u is odd.

Proof: Suppose thata∈ J+(q,n,u)∩J−

(q,n,u). Then there exist 0≤ i ≤ uqm/2−1 and 1≤ j ≤ uqm/2 suchthat

a∈Cn+i =Cn− j ,

which is equivalent to(2i +1)+(2 j −1)qℓ ≡ 0 (mod n)

for some 1≤ ℓ≤m−1. Observe that 2i+1 and 2j−1 are odd integers such that 1≤ 2i+1,2 j−1≤uqm−1.Then the desired results follow from the proof of Proposition 13.

Theorem 25. Let m≥ 5 be an odd integer, q even, andδ = uqm+1

2 /2+1. Let C be the primitive BCH codewith generator polynomial g(x) given by(3). Then the LCD codeC has length n= qm−1, dimension

k= qm−1− (uqm+1

2 −qu2)m,

and minimum distance d≥ 2δ−1.

Proof: The desired conclusion follows from Theorem 23, Proposition 24, and the BCH bound.

Example 9. 1) When(q,m,u) = (2,7,1), the codeC has parameters[127,29,37].2) Let (q,m)= (4,5). When u∈ {1,2,3}, the codeC has parameters[1023,723,d≥ 65], [1023,463,d≥

129], and [1023,243,d≥ 193], respectively.

B. Parameters ofC when m is evenIt has been seen from Proposition 20 thatCn+i 6=Cn+ j if and only if C2i+1 6=C2 j+1 (resp.Cn−i 6=Cn− j

if and only if C2i−1 6=C2 j−1). Let J be the set of integers that are not coset leaders, which are given by

16

(11). Note that 1≤ 2 j +1≤ uqm−1 if 0 ≤ j ≤ uqm/2−1 and 1≤ 2 j −1≤ uqm−1 if 1 ≤ j ≤ uqm/2. Itthen follows that

|J+(q,n,u)|= |J−(q,n,u)|= θ1m+θ2m/2+θ3m, (20)

whereθ1 = |{1≤ j ≤ uqm−1 : j is an odd coset leader and|C j |= m}|, (21)

θ2 = |{1≤ j ≤ uqm−1 : j is an odd coset leader and|C j |= m/2}|, (22)

andθ3 = |{1≤ j ≤ uqm−1 : j ∈ J is odd, the coset leader ofC j is even}|. (23)

Lemma 26. Let q be even, m even, and n= qm−1. Then we have the following.1)

θ1 =

{uqm/2−u(u−2)/4−u/2, if u is even;

uqm/2− (u−1)2/4− (u−1)/2, if u is odd.

2)

θ2 =

{u/2, if u is even;

(u−1)/2, if u is odd.

3)

θ3 =

{u(u−2)/8, if u is even;

(u2−1)/8, if u is odd.

Proof: Notice thatq is even. It then follows from Proposition 7, (21), and (22) that

θ1 = uqm/2−θ2−|{ j ∈ J : j is odd}|.

By (11), it is easy to see that

|{ j ∈ J : j is odd}| = |{ jmqm+ j0 : j0+1≤ jm≤ u−1,1≤ odd j0 ≤ u−1}|

=

{u(u−2)/4, if u is even,(u−1)2/4, if u is odd.

In addition, it follows from Proposition 7 that

θ2 = |{v(qm+1) : 1≤ odd v≤ u−1|=

{u/2, if u is even;

(u−1)/2, if u is odd.

Then we get the conclusions onθ1 andθ2.Let CL be the set of coset leaders ofC j , where j ∈ J. It follows from the proof of Proposition 7 that

CL = { j0qm+ jm : j0+1≤ jm ≤ u−1,1≤ j0 ≤ u−1}.

Then we can deduce from (23) that

θ3 = |{ j0qm+ jm : j0+1≤ even jm ≤ u−1,1≤ odd j0 ≤ u−1}|.

It can be easily verified that

θ3 =

{u(u−2)/8, if u is even;

(u2−1)/8, if u is odd.

This completes the proof.The following proposition follows from Lemma 26 and (20) directly.

17

Proposition 27. Let m≥ 2 be even. Then

|J+(q,n,u)|= |J−(q,n,u)|=

(uqm−u2/4

)m2 , if u is even;(

uqm− (u−1)2/4)

m2 , if u is odd.

Theorem 28. Let m≥ 2 be an even integer, q even, andδ = uqm2 /2+1.

1) If u is even, thenC(q,n,δ, n+12 ) and C(q,n,δ, n+1

2 −(δ−1)) both have length n= qm−1, dimension

k= qm−1−(

uqm2 −u2/4

)m2,

and minimum distance d≥ δ.2) If u is odd, thenC(q,n,δ, n+1

2 ) and C(q,n,δ, n+12 −(δ−1)) both have lenth n= qm−1, dimension

k= qm−1−(

uqm2 − (u−1)2/4

)m2,

and minimum distance d≥ δ.

Proof: The desired conclusions follow from Proposition 27 and the BCH bound directly.

Example 10. 1) When(q,m,u) = (2,6,1), the codeC(q,n,δ, n+12 ) ( or C(q,n,δ, n+1

2 −(δ−1))) has parameters[63,39,9], which are the best parameters for linear codes according tothe Database and the bestpossible cyclic codes according to [6, p. 260].

2) Let (q,m) = (4,4). When u∈ {1,2,3}, the codeC(q,n,δ, n+12 ) ( or C(q,n,δ, n+1

2 −(δ−1))) has parameters[255,223,d ≥ 9], [255,193,d ≥ 17], and [255,161,d ≥ 25], respectively.

The following conclusion will be employed to investigate the parameters of the codeC whenm≥ 2 iseven.

Proposition 29. For even m≥ 2, we have

J+(q,n,u)∩ J−(q,n,u) =⋃

j∈JE

Cn−( j+1)/2,

where the union is disjoint and

JE = { jmqm+(q−1)(qm−1+qm−2+ · · ·+q)+ j0 : 0≤ even jm ≤ u−1 and q−u≤ odd j0 ≤ q−1}.

Moreover,

|J+(q,n,u)∩ J−(q,n,u)|=

{u2m/4, if u is even;

(u+1)2m/4, if u is odd.

Proof: By Proposition 15, the proof of this proposition is very similar to that of Proposition 24 andis omitted.

Theorem 30. Let m≥ 2 be an even integer, q even, andδ = uqm2 /2+1. Let C be the primitive BCH code

with generator polynomial g(x) given by(3).1) If u is even, then the LCD codeC has length n= qm−1, dimension

k= qm−1− (uqm2 −u2/2)m,

and minimum distance d≥ 2δ−1.2) If u is odd, then the LCD codeC has length n= qm−1, dimension

k= qm−1− (uqm2 − (u2+1)/2)m,

and minimum distance d≥ 2δ−1.

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Proof: The desired conclusion follows from Theorem 28, Proposition 29, and the BCH bound.

Example 11. 1) When(q,m,u) = (2,4,1), the codeC has parameters[15,3,5], which are the bestpossible parameters for cyclic codes [6, pp. 247].

2) Let (q,m) = (4,4). When u∈ {1,2,3}, the codeC has parameters[255,195,17], [255,135,d≥ 33],and [255,83,d ≥ 49], respectively.

Corollary 31. When u= 1 and δ = qm2 /2+ 1, the true minimum distance of the codeC presented in

Theorem 30 is equal to2δ−1.

Proof: Note thatb= n+12 −δ+1. It is easy to check that(2δ−1) | gcd(n,b−1) in this case. The

desired result then follows from Corollary 3.C. Parameters ofC with designed distance qt −1, where1≤ t ≤ mWhenq is even, the dimension and minimum distance of the LCD codeC are described in the following

theorem ifC has designed distance 2δ = qt −1 for an integert with 1≤ t ≤ m.

Theorem 32. Let q be even, m≥ 2 and m6= 3. LetC be the primitive BCH code with generator polynomialg(x) given in(3) with designed distance2δ−1= qt −1, where1≤ t ≤ m. ThenC is an LCD cyclic codeand has length n= qm−1, dimension

k=

{qm−1− (q

m+12 −q)m if m≥ 5 is odd and t= m+1

2 ,

qm−1− (qt −2)m otherwise,

and minimum distance d≥ qt −1.

Proof: For evenq, it is known thatCn+i 6=Cn+ j if and only if C2i+1 6=C2 j+1 (resp.Cn−i 6=Cn− j ifand only if C2i−1 6= C2 j−1). Note thatδ = qt

2 . It is clear that 1≤ 2 j +1 ≤ qt −3 if 0 ≤ j ≤ δ−2 and1≤ 2 j −1≤ qt −3 if 1 ≤ j ≤ δ−1. It then follows from the proofs of Propositions 13 and 15 that

∣∣∣( ⋃

1≤ j≤δ−1

Cn+ j

)⋂( ⋃

1≤ j≤δ−1

Cn− j

)∣∣∣={(q

2 −1)2m if m≥ 5 is odd andt = m+12 ,

0 otherwise.

The generator polynomial of the codeC is g(q,n,2δ−1, n+12 −(δ−1))(x). By Lemma 4, every odd integeri

in the range 1≤ i ≤ qt −3 (1≤ t ≤ m) is a coset leader whenq is even. We then have

deg(g(q,n,2δ−1, n+12 −(δ−1))(x)) = (qt −2)m−

∣∣∣( ⋃

1≤ j≤δ−1

Cn+ j

)⋂( ⋃

1≤ j≤δ−1

Cn− j

)∣∣∣.

Thus the dimension is straightforward and the desired conclusion on the minimum distance ofC followsfrom the BCH bound.

It should be remarked that the minimum distance of the codeC given in Theorem 32 may be largerthanqt −1.

Example 12. 1) Let (q,m)= (2,7). When t∈{2,3,4}, the codeC has parameters[127,113,5], [127,85,11]and [127,29,37] with designed distance3, 7, and 15, respectively.

2) Let (q,m) = (2,6). When t∈ {2,3}, the codeC has parameters[63,51,3] and [63,27,7].

VII. PARAMETERS OF THELCD CYCLIC CODESC(q,n,2δ,n−δ+1), WHERE δ = uqm

Recently, a class of LCD cyclic codesC(q,n,2δ,n−δ+1) with generator polynomialsg(q,n,2δ,n−δ+1)(x) werepresented and their dimensions were determined whenδ ≤ T +1 [13], whereT is given by (5). In thissection, we continue to investigate the parameters of the codesC(q,n,2δ,n−δ+1) for δ = uqm.

Let 1≤ u≤ q−1 be an integer. Define

J+(q,n,u) =⋃

1≤ j≤uqm

Cj and J−(q,n,u) =⋃

1≤ j≤uqm

Cn− j .

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The following proposition will play a key role in presentingparameters of the LCD cyclic codesC(q,n,2δ,n−δ+1).

Proposition 33. Assume that n= qm−1 and 1≤ u≤ q−1.1) If m≥ 5 is odd, then

|J+(q,n,u)|= |J−(q,n,u)|= (uqm−1−u2+u)(q−1)m

and|J+(q,n,u)∩ J−(q,n,u)|= 2u2(q−1)m.

2) If m≥ 2 is even, then

|J+(q,n,u)|= |J−(q,n,u)|= uqm−1(q−1)m−(u−1)2

2m

and|J+(q,n,u)∩J−(q,n,u)|= u2m.

Proof: It can be easily deduced from Propositions 5, 7, 13, and 15.

Theorem 34. Let δ = uqm+1, where1≤ u≤ q−1.1) When m≥ 5 is an odd integer, the LCD cyclic codeC(q,n,2δ,n−δ+1) has length n= qm−1, dimension

k= qm−2−2(uqm−1

2 −2u2+u)(q−1)m,

and minimum distance d≥ 2δ. In addition, the generator polynomial is given by

g(q,n,2δ,n−δ+1)(x) = ∏1≤ j≤uq

m+12

q∤ j , j 6∈J1∪J2∪JO

mj(x)mn− j(x).

2) When m≥ 2 is an even integer, the codeC(q,n,2δ,n−δ+1) has length n= qm−1, dimension

k= qm−2−2uqm2 −1(q−1)m+(u−1)2m+u2m,

and minimum distance d≥ 2δ. In addition, the generator polynomial is given by

g(q,n,2δ,n−δ+1)(x) = ∏1≤ j≤uq

m2

q∤ j , j 6∈J

mj(x) ∏1≤ j≤uq

m2

q∤ j , j 6∈J∪JE

mn− j(x).

Proof: It follows from Proposition 33 and the BCH bound immediately.

Example 13. 1) When(q,m,u) = (2,4,1), the codeC(q,n,2δ,n−δ+1) has parameters[15,2,10], which isan optimal code according to the Database.

2) When(q,m,u) = (4,2,1), the codeC(q,n,2δ,n−δ+1) has parameters[15,4,10], which is an optimalcode according to the Database.

3) Let (q,m) = (3,5). When u∈ {1,2}, the codeC(q,n,2δ,n−δ+1) has parameters[2184,1457,d ≥ 164]and [2186,841,d≥ 326], respectively.

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VIII. C ONCLUDING REMARKS

The main contributions of this paper are the following:1) The characterization of the coset leaders in the range 1≤ j ≤ (q−1)qm and the cardinalities of their

cyclotomic cosets.2) The construction of the class of LCD cyclic codesC and the analysis of their parameters when

δ = uqm+1 if q is odd andδ = uqm/2+1 if q is even for 1≤ u≤ q−1.3) The analysis of the parameters of the LCD cyclic codeC when it has designed distanceqt −1,

where 1≤ t ≤ m.4) The analysis of the parameters of the LCD cyclic codesC(q,n,2δ,n−δ+1) whenδ = uqm.The dimensions of all these codes were presented and lower bounds on their minimum distances were

derived from the BCH bound whenδ = uqm+1, uqm/2+1, or qt −1 for 1≤ u≤ q−1 and 1≤ t ≤ m.The minimum distances of some codes were also determined.

Since we characterized the coset leaders in the range 1≤ j ≤ (q−1)qm and presented Propositions 13,15, 24, and 29, the dimensions and lower bounds on the minimumdistances of the codesC can be settledfor 1≤ δ ≤ uqm+1 if q is odd andδ = uqm/2+1 if q is even. It should be mentioned that the parametersof several classes of LCD cyclic codes of lengthsqm−1

q−1 and qℓ+1 were analysed in [9]. The technique

employed in this paper may be applied to investigate the parameters of the codesC whenn= qm−1q−1 and

qℓ+1. Experimental data show that many good codes do exist and giving a further study is worthwhile.The codes studied in this paper are different from those in [7] and [8], which are not LCD codes. They

are also different from those in [13], as the former is generated by the polynomialg(x) given in (4), whilethe latter has generator polynomialg(q,n,2δ,n−δ+1)(x). The codes of this paper are reversible primitive BCHcodes, while the reversible BCH codes presented in [9] are not primitive. The lines above briefly informthe reader the major differences between this paper and references [7], [8], [9] and [13].

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