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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 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media New York. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 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 Cryptogr. Commun. 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 prin- cipal, 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 nota- tions 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 Cryptogr. Commun. 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.
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