New Developments in Q-Polynomial Codes

New developments in q-polynomial codes

Chuan Lv, Tongjiang Yan & Guozhen Xiao

Cryptography and Communications Discrete Structures, Boolean Functions and Sequences

ISSN 1936-2447

Cryptogr. Commun. DOI 10.1007/s12095-015-0147-4

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1 23 Author's personal copy

Cryptogr. Commun. DOI 10.1007/s12095-015-0147-4

New developments in q-polynomial codes

Chuan Lv1 · Tongjiang Yan2 · Guozhen Xiao1

Received: 19 March 2015 / Accepted: 12 June 2015 © Springer Science+Business Media New York 2015

Abstract Cyclic codes are prominently used in electronics and communication engineer- ing. A new view on cyclic codes based on q-polynomials has been introduced by Ding and Ling. This paper is concerned with new developments in cyclic codes from q-polynomials. The properties of the q-polynomial codes and the fundamental relations between q- polynomial codes and generator polynomials are proposed. Then a new design of t-error correcting codes is introduced. Several constructions of new q-polynomial codes from old ones are also presented.

Keywords Cyclic codes · Linearized polynomial · q-Polynomial · Trace function

Mathematics Subject Classification (2010) 94B15 · 94B05 · 05B50

1 Introduction

Let q beapowerofaprime,alinear[n, k, d; q] code is a k-dimensional subspace of GF(q) with minimum nonzero (Hamming) weight d. A linear [n, k] code C over GF(q) is called cyclic if (c0,c1, ··· ,cn−1) ∈ C implies (cn−1,c1,c2, ··· ,cn−2) ∈ C. By identifying any

The work is supported by the National Natural Science Foundations of China (No.61170319), the Shandong Provincial Natural Science Foundation of China(No. ZR2014FQ005) and the Fundamental Research Funds for the Central Universities of China(No. 15CX02081A).

Chuan Lv [email protected] Tongjiang Yan [email protected]

1 State Key Laboratory of ISN, Xidian University, Xi’an 710071, People’s Republic of China

2 School of Sciences, China University of Petroleum, Qingdao 366580, People’s Republic of China Author's personal copy

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n vector (c0,c1, ··· ,cn−1) ∈ GF(q) with a polynomial

2 n−1 n c0 + c1x + c2x +···+cn−1x ∈ GF(q)[x]/(x − 1), any code C of length n over GF(q) corresponds to a subset of the quotient ring GF(q)[x]/(xn − 1). A linear code C is cyclic if and only if the corresponding subset is an ideal of the ring GF(q)[x]/(xn − 1). Noting that every ideal of GF(q)[x]/(xn − 1) is principal, any cyclic code C can be expressed as C = (g(x)),whereg(x) is monic and has the smallest degree among all the generators of C.Theng(x) is unique and called the generator polynomial, and h(x) = (xn − 1)/g(x) is referred to as the parity-check polynomial of C. Cyclic codes have applications in storage and communication systems because they have efficient encoding and decoding algorithms [2, 3, 5]. Cyclic codes have been studied for decades and a lot of progress has been made. Three approaches are generally used in the design and analysis of cyclic codes, while they are based on generator matrices, generator polynomials and idempotents respectively. Ding and Ling introduced a new approach to those of cyclic codes based on q-polynomials and constructed some new cyclic codes[4]. The main concerns of this paper are to build the fundamental theory of the q-polynomial codes and to give some constructions of the cyclic codes from the q-polynomial approach. The remainder parts of this paper are organized as follows. Section 2 gives some notations for this paper, some properties of q-polynomials and several lemmas. The relation between generator polynomials and q-polynomial codes is discussed in Section 3. In Sec- tion 4,anewdesignoft-error correcting codes which are called q-BCH codes is introduced and the dimensions of such codes are considered. Several constructions of cyclic codes from original q-polynomial codes are presented in Section 5. Section 6 summarizes this paper.

2 Preliminaries

Throughout this paper, we adopt the following notations: • p is a prime. • q is a positive power of p. • n is a positive integer and is used to denote the length of a cyclic code over GF(q). • r = qn. • t s | Trqt /qs (x) is the trace function from GF(q ) to GF(q ),wheres t. • A B is the direct sum of the two subspaces A and B. • DimA is the dimension of the space A.

A q-polynomial, or a linearized polynomial over GF(q) is a polynomial of the form = h qi L(x) i=0 lix with all coefficients li in GF(q) and h being a nonnegative integer. The = h i = h qi polynomials l(x) i=0 lix and L(x) i=0 lix over GF(q) are called q-associates of each other. More specially, l(x) is the conventional q-associate of L(x) and L(x) is the linearized q-associate of l(x).Giventwoq-polynomials L1(x) and L2(x), the symbolic multiplication of them is defined by L(x) = L1(x) L2(x) = L1(L2(x)) = L2(L1(x)). Both L1(x) and L2(x) symbolically divide L(x). In the following we give an definition and some lemmas which are needed in this paper.

Definition 1 [8]LetL(x) be a nonzero q-polynomial over GF(q). A root ζ of L(x) is called a q-primitive root over GF(q) if it is not a root of any nonzero q-polynomial over GF(q) of lower degree. Author's personal copy

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Lemma 1 [8] Let L1(x) and L2(x) be q-polynomials over GF(q) with conventional = = q-associates l1(x) and l2(x), respectively. Then l(x) l1(x)l2(x) and L(x) L1(x) L2(x) are q-associates of each other.

Lemma 2 [8] Let L1(x) and L(x) be q-polynomials over GF(q) with conventional q- associates l1(x) and l(x), respectively. Then the following properties are equivalent: (i) L1(x) symbolically divides L(x); (ii) L1(x) divides L(x); (iii) l1(x) divides l(x).

Lemma 3 [9] Let C be a cyclic code with generator polynomial g(x) and parity-check polynomial h(x) = (xn − 1)/g(x). The dual code C⊥ is cyclic and has the generator polynomial ⊥ − g (x) = xdegh(x)h(x 1).

n n Lemma 4 [8] Let x −1 = f1(x)f2(x) ···fm(x) be the decomposition of x −1 into monic irreducible factors over GF(q).If(n, q) = 1, there are no multiple factors.

Cyclic code is called maximal cyclic code if it is generated by an irreducible factor n fi(x), and is called irreducible cyclic code if it is generated by x − 1/f i(x)[8]. Any cyclic code can be represented as a direct sum of irreducible cyclic codes, which implies that the number of the cyclic codes depends on the factorization of xn − 1inGF(q).If(q, n) = 1, the number is equal to 2m by Lemma 4.

3 Generator polynomials and q-polynomial codes

Let q be a prime power, n be a positive integer and r = qn and λ be an element of GF(r)∗. A q-polynomial code with the check element λ is defined in [4]by n−1 n qi Cλ = (c0,c1, ··· ,cn−1) ∈ GF(q) : C(λ) = 0, where C(x) = cix . (1) i=0

Ding and Ling studied the properties of Cλ and proved the following:

Proposition 1 [4] Every cyclic code of length n over GF(q) can be expressed as the code n Cλ forsomeelementλ ∈ GF(q ), and is thus a q-polynomial code.

A constructive method basing on initial seed sequences is employed to find λ of Cλ for a given cyclic code in [4]. Hereafter we give the relation between the generator polynomial of the cyclic code and the check element λ of the corresponding q-polynomial code Cλ,then give another proof of Proposition 1. By the Normal Basis Theorem [7], GF(r) has a normal n−1 basis {α, αq , ··· ,αq } over GF(q),whereα ∈ GF(r)∗ and α is called a normal element of GF(r) over GF(q).Thenλ has a unique expression of the form

n−1 qi λ = λiα , (2) i=0 where each λi ∈ GF(q). Since the normal element α is a q-primitive root of q-polynomial n xq −x, there exists no q-polynomial L(x) with the degree less than qn such that L(α) = 0. Author's personal copy

Cryptogr. Commun. = n−1 qi = Let (x) i=0 λix ,thenλ (α).Theq-polynomial code can be denoted by n−1 n qi Cλ = (c0,c1, ··· ,cn−1) ∈ GF(q) : C((α)) = C(α) (α) = 0, where C(x) = ci x . (3) i=0 In the following we give Theorems 1 and 2 to illustrate the one-to-one correspondence between q-polynomial codes and cyclic codes.

Theorem 1 Let the notations be the same as above, a q-polynomial code Cλ with the check = n−1 qi = n − n − elementλ i=0 λiα is the cyclic code C ((x 1)/gcd(x 1, λ(x))),where = n−1 i λ(x) i=0 λix . ··· ∈ = n−1 qi Proof If (c0,c1, ,cn−1) Cλ, we denote C(x) i=0 cix .Sinceα is a normal qn qn element, x − x is the monic q-polynomial of the least degree with root α,i.e.α − qn α = 0. Therefore C(α) (α) = 0in(3) implies that x − x symbolically divides C(x) (x). By Lemma 2, xn − 1dividesc(x)λ(x),i.e.c(x)λ(x) ≡ 0mod(xn − 1), where c(x) and λ(x) are conventional q-associates of C(x) and (x) respectively. Therefore n n n gcd(x − 1, λ(x)) is the parity-check polynomial of Cλ and (x − 1)/gcd(x − 1, λ(x)) is the generator polynomial.

Theorem 2 A cyclic code of length n over GF(q) with the parity-check polynomial λ(x) = h λ0 + λ1x +···+λhx (h ≤ n − 1) is a q-polynomial code with the check element λ = q qh λ0α + λ1α +···+λhα .

Proof Let (x) be the linearized q-polynomial of λ(x).SinceC = (g(x)) is a principal ideal of GF(q)[x]/(xn − 1), one can see that g(x) divides any c(x) ∈ C,where g(x) = (xn − 1)/gcd(xn − 1, λ(x)). Therefore we have g(x) = (xn − 1)/gcd(xn − 1, λ(x)) divides c(x), which implies that xn − 1dividesc(x)λ(x).ByLemmas1and2, n xq − x divides C(x) (x) = C((x)),i.e.C((α)) = C(λ) = 0, where C(x) and c(x) are q-associates with each other. Both C(x) and c(x) correspond to the codeword (c0,c1, ··· ,cn−1).

Remark 1 Theorem 1 was first introduced and proved by Ding and Ling (Theorem 4.6 in [4]), while here our proof is different and simple.

Remark 2 Proposition 1 can be derived directly from Theorem 2 as a corollary, i.e. we give a different and very short proof of Proposition 1. Theorem 2 also answers the first question proposed on page 10 of [4].

Remark 3 In [4], initial seed sequences were used to find the check element λ, here we can get it directly from the parity-check polynomial and present the explicit expression of λ. Theorems 1 and 2 make the relation between the classical approach and the q-polynomial approach to cyclic codes much clearer.

Let g(x) be the generator polynomial of the cyclic code Cλ.If(q, n) = 1, g(x) has only simple roots. Therefore G(x) has only simple roots, where G(x) is the linearized q- polynomial of g(x). Theorem 3.70 in [8] reveals the existence of q-primitive roots, and the number of the q-primitive roots Ng of G(x) is equal to q (g(x)),whereq (g(x)) is the number of polynomials of degree less than n and relatively prime to g(x). We can see that Author's personal copy

Cryptogr. Commun. two check elements λ and λ lead to the same q-polynomial codes if both λ and λ are q- primitive roots of the q-polynomial. Combining the discussion above with Lemma 3.69 in [8] leads to = = n−1 qi = n−1 i = n − Theorem 3 Suppose λ (α) i=0 λiα , λ(x) i=0 λix and l(x) gcd(x 1, λ(x)) whose degree is m. The number of check elements in GF(r)∗ whichleadtothesame q-polynomial code Cλ is equal to

m −m1 −ms q (l(x)) = q (1 − q ) ···(1 − q ), where the mi’s are the degrees of the distinct monic irreducible polynomials appearing in the canonical factorization of l(x) in GF(q)[x].

i If λ is a q-primitive root of a q-polynomial L(x), λq and bλ (where b ∈ GF(q)∗)are i also q-primitive roots of L(x). Therefore λ = αq or λ = bα is an sufficient condition = such that Cλ C λ (Theorem 4.9 in [4]). We point out that the condition is not necessary. i For certain element λ, Theorem 3 indicates the existence of λ which is not the form λq = = 6 + 5 + 4 + 3 + 2 + or bλ such that Cλ C λ. For example, let λ(x) x x 2x x 2x 2x and 6 5 4 3 2 λ = α2 + α2 + 2α2 + α2 + 2α2 + 2α2,whereα is a normal element in GF(310). We have that gcd(x10 − 1, λ(x)) = (x4 + 2x3 + x2 + 2x + 1)(x + 2) is the canonical factorization over GF(3). By Theorem 3, the size of q-primitive roots set containing λ is i 35(1 − 3−4)(1 − 3−1) = 160. While the number of the elements of the form λ3 and bλ is 11, which is less than 160. The relation between the check element of a q-polynomial code and its generator polynomial is as follows.

Proposition 2 [4] Let γ be a generator of GF(r)∗, and let λ = γ s for some positive integer s.Letls denote the size of the q-cyclotomic coset modulo r − 1 containing s.LetHλ denote the subspace of GF(r) over GF(q) spanned by the set

l −1 {λ, λq , ··· ,λq s }.

Define Gλ(x) = (x − z). z∈Hλ

Then Gλ(x) is a q-polynomial and its conventional q-associate gλ(x) is the generator polynomial of the code Cλ.

Proof We give another proof which is different from the one in [4]. Let Gλ(x) be the monic q-polynomial of the least degree that annihilates λ,andgλ(x) be the conven- qi tional q-associate of Gλ(x). Gλ(x) is a q-polynomial implies that λ are also roots of Gλ(x). Therefore all the roots of Gλ(x) form a subspace of GF(r) which is spanned by l −1 {λ, λq , ··· ,λq s }.Thenwehave

Gλ(x) = (x − z). z∈Hλ

Since Gλ(x) is of the least degree, Gλ(x) divides any q-polynomial C(x) that annihilates λ. By Lemma 2, gλ(x) is the polynomial of the least degree that divides c(x),wherec(x) and C(x) are q-associates of each other, thus gλ(x) is the generator polynomial of Cλ. Author's personal copy

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4 Properties of q-BCH codes

∗ Let S be any subset of GF(r) .Wedefinetheq-polynomial code CS with check set S by n−1 n qi CS ={(c0,c1, ··· ,cn−1) ∈ GF(q) : C(λ) = 0∀λ ∈ S, where C(x) = cix }. (4) i=0

= Note that CS λ∈S Cλ . CS is also a cyclic code.

k k +1 k +d−2 Definition 2 Let R ={γ 0 ,γ 0 , ··· ,γ 0 },wherek0 and d are two integers with n 0 ≤ k0 ≤ q − 2and2≤ d ≤ n.Theq-polynomial code CR with check set R is called a q-BCH code.

Although the q-BCH code CR defined here is different from the ordinarily defined BCH codes [1, 6], the following discussion shows their properties are similar.

Theorem 4 The minimum distance of the q-BCH code CR in Definition 2 is at least d.

Proof Let c = (c0,c1, ··· ,cn−1) be any codeword in CR. By the definition of CR,we immediately have + + − C(γ k0 ) = C(γ k0 1) =···=C(γ k0 d 2) = 0. This leads to ⎛ ⎞ n−1 ⎛ ⎞ ⎛ ⎞ k0 k0q ··· k0q γ γ γ c0 0 ⎜ k k q k qn−1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ γ 1 γ 1 ··· γ 1 ⎟ ⎜ c1 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ (5) ⎜ . . . . ⎟ ⎝ . ⎠ ⎝ . ⎠ ⎝ . . .. . ⎠ . . n−1 γ kd−2 γ kd−2q ··· γ kd−2q cn−1 0 where ki = k0 + i for any 1 ≤ i ≤ d − 2. Suppose that c has weight ω with ω ≤ d − 1, i.e. ci = 0 if and only if i ∈{a0,a1, ··· ,aω−1}.By(5), we have

⎛ a a a − ⎞ ⎛ ⎞ ⎛ ⎞ k0q 0 k0q 1 ··· k0q ω 1 γ γ γ ca0 0 a a a − ⎜ k1q 1 k1q 1 ··· k1q ω 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ γ γ γ ⎟ ⎜ ca1 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ . . . . ⎠ ⎝ . ⎠ ⎝ . ⎠ . (6) ...... a a a − kω−1q 0 kω−1q 1 kω−1q ω 1 γ γ ··· γ caω−1 0

qai Set βi = γ for each 0 ≤ i ≤ ω − 1. It is clear that βi = βj for i = j.Then(6) becomes ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ k0 k0 k0 β β ··· β − c 0 ⎜ 0 1 ω 1 ⎟ a0 k1 k1 ··· k1 ⎜ ⎟ ⎜ ⎟ ⎜ β β β − ⎟ ⎜ ca1 ⎟ ⎜ 0 ⎟ ⎜ 0 1 ω 1 ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ . . . . ⎟ ⎝ . ⎠ ⎝ . ⎠ . (7) ⎝ . . .. . ⎠ . . k − k − k − ω 1 ω 1 ··· ω 1 ca − 0 β0 β1 βω−1 ω 1

Note that k0,k1, ··· ,kd−1 are d consecutive integers. According to Lemma 3.51 in [8], it is easy to verify that the determinant of the matrix on the left of (7) is not zero. Thus we = ≤ ≤ − = have cai 0 for each 1 i ω 1. This is a contradiction with the fact that cai 0for each 1 ≤ i ≤ ω − 1. The proof is finished. Author's personal copy

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For example, let (q, n) = (2, 6),andletγ be a generator of GF(r)∗ with γ 6 + γ 5 + γ 3 + γ 2 + 1 = 0. Define R ={γ 27,γ28}. Theorem 4 indicates that the minimal distance of CR is at least 3. Actually, CR is a [6, 2, 3] cyclic code over GF(2) with generator polynomial x4 + x2 + 1.

− = Theorem 5 Let lki be the size of the q-cyclotomic coset modulo r 1 containing ki, i ··· − − d−2 0, 1, ,d 2. The dimension of CR in Definition 2 is no less than n i=0 lki .

= ki ≤ ≤ − Proof Let ρi γ for each 0 i d 2. Since lki is the size of the q-cyclotomic coset − modulo r 1 containing ki, similar to that was defined in Proposition 2, we denote Hρi as l { q ··· q ki −1} the subspace of GF(r) over GF(q) spanned by ρi,ρi , ,ρi . Then the dimension of − H is no larger than d 2 l . By Proposition 2, the degree of the generator polynomial i ρi i=0 ki d−2 of CR is no larger than i=0 lki , then the result is immediate.

Theorem 5 shows the relation between the dimension of q-BCH codes and the q- cyclotomic cosets of each ki. In order to construct a cyclic code with large dimension and given minimum distance, it is preferable to choose ki such that each size lki is small. We note that if a normal element α is contained in the check set R, the dimension of the q-BCH code CR is 0, which is the worst case.

5 Constructions of q-polynomial codes from original ones

E Let the q-polynomial code Cλ be denoted in (1), we define the code Cλ as follows n−1 E ={ ··· ∈ n : ∈ = qi } Cλ (c0,c1, ,cn−1) GF(q) C(λ) GF(q), where C(x) cix . (8) i=0 E ⊆ E It is easy to see that Cλ is cyclic and Cλ Cλ . If there exists a codeword ··· = E + (c0,c1, ,cn−1) such that C(λ) 0, we have that Dim(Cλ ) is equal to Dim(Cλ) 1. The existence of such codeword lays on the value of the trace function Trqn/q (λ),andthe E code Cλ is expressed in the following theorem.

Theorem 6 For n such that gcd(n, q) = 1, we have E = = (i) Cλ Cλ,ifTrqn/q (λ) 0; E = = = (ii) Cλ Cλq −λ Cλ 1,ifTrqn/q (λ) 0,where1 is the repetition code over GF(q),i.e. 1 ={(k, k, ··· ,k),k ∈ GF(q)}.

Proof For the first part, we suppose that there exists a codeword (c0,c1, ··· ,cn−1) such i i that C(λ) = k, k ∈ GF(q)∗. Then for i ∈{0, 1, ··· ,n− 1},wehaveC(λq ) = C(λ)q = C(λ). Therefore

q qn−1 q qn−1 C(Trqn/q (λ)) = C(λ + λ +···+λ ) = C(λ) + C(λ ) +···+C(λ ) = nC(λ) = nk = 0.

Since Trqn/q (λ) = 0,wehaveC(Trqn/q (λ)) = 0, which is a contradiction. Then we prove the second part. Let (c0,c1, ··· ,cn−1) be any codeword in Cλ and let C(x) be its corresponding q-polynomial. Since Trqn/q (x) is a q-polynomial with the Author's personal copy

Cryptogr. Commun. codeword (1, 1, ··· , 1), C(x) + kTrqn/q (x) is also a q-polynomial, k ∈ GF(q).Since

C(λ) + kTrqn/q (λ) = kTrqn/q (λ) ∈ GF(q), + + ··· + ∈ E E = we have (c0 k,c1 k, ,cn−1 k) Cλ . Therefore we get Cλ Cλ 1. ··· ∈ E q − = For any (c0,c1, ,cn−1) Cλ , C(λ) is in GF(q), therefore we have C(λ λ) q q C(λ ) − C(λ) = C (λ) − C(λ) = 0, which leads to (c0,c1, ··· ,cn−1) ∈ Cλq −λ.Onthe q contrary, for any (c0,c1, ··· ,cn−1) ∈ Cλq −λ, C(λ − λ) = 0 implies C(λ) ∈ GF(q),sowe ··· ∈ E E = have (c0,c1, ,cn−1) Cλ . Therefore we reach Cλ Cλq −λ.

If Trqn/qt (λ) = 0 for an integer t|n, we define the expansion of Cλ as n−1 i Et ={ ··· ∈ n : ∈ t = q } Cλ (c0,c1, ,cn−1) GF(q) C(λ) GF(q ), where C(x) cix . (9) i=0 By Lemmas 1 and 2, similar discussion as the proof of Theorem 6 leads to the following theorem.

Theorem 7 For n such that gcd(n, q) = 1, we have

Et = = (i) Cλ Cλ,ifTrqn/qt (λ) 0; Et = = = (ii) Cλ Cλqt −λ Cλ D,ifTrqn/qt (λ) 0, where the code D is the cyclic code generated by the codeword corresponding to the q-polynomial Trqn/qt (x).

Theorems 6 and 7 propose two methods of expanding a given q-polynomial code by adding (direct sum) a cyclic code. It should be mentioned that other q-polynomials can also be used to expanding the code Cλ besides the trace functions. More generally, if there exists = n−1 qi = E(x) a q-polynomial E(x) i=0 eix such that E(λ) 0, we can expand Cλ to Cλ by adding (direct sum) a cyclic code generated by the codeword (e0,e1, ··· ,en−1). It is not difficult to show that the expanded code corresponding to E(x) can also be expressed as n−1 E(x) ={ ··· ∈ n : ∈ = qi } Cλ (c0,c1, ,cn−1) GF(q) C(λ) ME, where C(x) cix , i=0 where the set ME is the smallest q-modulo containing E(λ). = n−1 qi = On the contrary, if there exists a q-polynomial F(x) i=0 fix such that F(λ) 0, i.e. (f0,f1, ··· ,fn−1) ∈ Cλ. We can expurgate Cλ by throwing away those codewords as (f0,f1, ··· ,fn−1), therefore the expurgated code can be expressed as n−1 n qi {(c0,c1, ··· ,cn−1) ∈ GF(q) : C/F(λ) = 0, where C(x) = cix }, i=0 where C/F(x) is the symbolical division of C(x) and F(x). This code is also a cyclic code constructed from the original code Cλ defined in (1). = n−1 qi ˆ = n−1 qi Let λ i=0 λiα . We denote the reciprocal of λ as λ i=0 λn−1−iα and denote the reciprocal of g(x) as g(x)ˆ = xdegg(x)g(x−1). Besides the above constructions proposed, the dual code and the reciprocal code of a cyclic code can also be obtained easily from the q-polynomial approach [9]. We express them as follows. = n−1 qi = Theorem 8 Let Cλ be the code denoted in (1) where λ i=0 λiα .Letλ(x) n−1 i i=0 λix , we have Author's personal copy

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⊥ = ˆ ˆ [ ] n − ˆ (i) Cλ (λ(x)),where(λ(x)) is the ideal of GF(q) x /(x 1) generated by λ(x); ˆ (ii) The reciprocal code of Cλ is Cλˆ ,whereλ is the reciprocal of λ in GF(r).

Proof For the first part: In the proof of Theorem 1, we proved that λ(x) is the parity-check ˆ polynomial of Cλ. By Lemma 3, λ(x) is the generator polynomial. ··· ∈ Then we prove the second part. By Theorems 1 and 2, (c0,c1, ,cn−1) Cλ n − = n−1 i if and only if x 1dividesc(x)λ(x),wherec(x) i=0 cix . Since the reciprocal of xn − 1 is itself, we have that xn − 1dividesc(x)ˆ λ(x)ˆ ,wherec(x)ˆ corresponds ˆ ˆ to the codeword (cn−1,cn−2, ··· ,c0) and λ(x) corresponds to the element λ. Therefore ··· (cn−1,cn−2, ,c0) is in Cλˆ . = n−1 qi ∈ R = Corollary 1 Let CR be the code denoted in (4). For any λ i=0 λiα ,letλ(x) n−1 i i=0 λix . Then we have ⊥ ˆ (i) CR = (lR(x)),wherelR(x) is the least common multiple of all λ(x) with λ ∈ R, and (lR(x)) is the ideal of GF(q)[x]/(xn − 1) generated by lR(x); ˆ ˆ (ii) The reciprocal code of CR is CRˆ ,whereR ={λ :∀λ ∈ R}.

6 Conclusion and final remark

The main results of this paper are the following: 1. Properties of check element of a q-polynomial and its relation with generator polynomial in Section 3. 2. Introduction of q-BCH codes and the properties of such codes in Section 4. 3. Constructions of cyclic codes from q-polynomial code originally defined in (1)in Section 5. Several properties of q-polynomial codes were presented in Section 3. Although Proposi- tions 1 and 2 are originally introduced in [4], we gave new proofs of these two propositions. In Section 4,theq-BCH code can be viewed as a new design of t-error correcting codes. We obtained that the minimum distance is at least d, and that its dimension is closely related to the q-cyclotomic coset modulo r − 1. Section 5 proposed several methods of constructing cyclic codes from a given code Cλ, which include expansion, expurgation, dual code and reciprocal code. The results in Sections 4 and 5 show that the q-polynomial approach is flex- ible and promising. It is an interesting problem to find more applications of this approach to the design and analysis of cyclic codes.

Acknowledgments The authors are very grateful to the editor and the anonymous reviewers for their valuable comments and suggestions to improve the quality of this paper.

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