AAECC (2011) 22:219–233 DOI 10.1007/s00200-011-0145-5 ORIGINAL PAPER
Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration Yonglin Cao
Received: 16 January 2010 / Revised: 6 May 2011 / Accepted: 9 May 2011 / Published online: 22 May 2011 © Springer-Verlag 2011
Abstract For R a Galois ring and m 1 , . . . , m l positive integers, a generalized quasicyclic (GQC) code over R of block lengths (m 1 , m 2 , . . . , m l ) and length li=1 m i is an R[x]-submodule of R[x]/(x m 1 − 1) × · · · × R[x]/(x m l − 1). Suppose m 1 , . . . , m l are all coprime to the characteristic of R and let {g1 , . . . , gt } be the set of all monic basic irreducible polynomials in the factorizations of x m i − 1 (1 ≤ i≤ l). Then the GQC codes over R of block lengths (m 1 , m 2 , . . . , m l ) and length li=1 m i are identified with G 1 × · · · × Gt , where G j is an R[x]/(g j )-submodule of (R[x]/(g j ))n j , where n j is the number of i for which g j appears in the factorization of x m i − 1 into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths. Keywords
Generalized quasi-cyclic code · Galois ring · Enumeration · Submodule
Mathematics Subject Classification (2000)
11T71 · 94B05
1 Introduction The notion of generalized quasi-cyclic (GQC) codes over finite fields was introduced by Siap and Kulhan [11] a few years ago, and some of the structural properties of such codes were studied by Esmaeili and Yari [4] more recently. As a natural generalization, we introduce
Y. Cao (B) School of Sciences, Shandong University of Technology, Zibo, Shandong 255091, People’s Republic of China e-mail:
[email protected]
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Definition 1.1 Let R be a commutative ring with identity and m 1 , m 2 , . . . , m l positive integers. Denote Ai = R[x]/(x m i − 1) for i = 1, . . . , l. Any R[x]-submodule of the quasi-cyclic (GQC)code R[x]-module A = A1 × A2 ×· · ·× Al is called a generalized over R of block lengths (m 1 , m 2 , . . . , m l ) and length li=1 m i . As noted in [11] for codes over finite fields, it is clear that if C is a GQC code over R of block lengths (m 1 , m 2 , . . . , m l ) and length li=1 m i with m = m 1 = m 2 = . . . = m l , then C is a quasi-cyclic (QC) code over R of length lm and index l. Further, if l = 1, then C is a cyclic code over R of length m. Notation For simplicity, every GQC code means a GQC code of block lengths (m 1 , m 2 , . . . , m l ) and length li=1 m i , and every QC code means a QC code of length lm and index l in Sects. 3–5 of this paper. Linear codes over rings have recently raised a great interest for their new role in algebraic coding theory and for successful application in combined coding and modulation. Compared to QC codes, an advantage of GQC codes is that one can construct GQC codes of any length; also, one can construct GQC codes having dimensions larger than those for the QC codes of the same length over finite fields (see [4]). When R = Fq is a finite field of cardinality q, Conan and Séguin [3] have investigated structural properties and enumeration of QC codes and enumeration of some subclasses of QC codes including the cyclic codes, the QC codes with a cyclic basis, the maximal and the irreducible ones over Fq . Ling and Solé [5–7] proposed to regard a QC code over Fq as a linear code over an auxiliary ring. According to their idea, that ring can be decomposed into a direct product of fields. They decomposed QC codes by the Chinese Remainder Theorem (CRT), or equivalently the MattsonSolomon transform, into products of shorter codes over larger alphabets. They enumerated all 1-generator codes and outlined a generalization to multi-generator codes. Séguin [10] has studied structural properties and enumeration of an important subclass of 1-generator QC codes over Fq , described an algorithm for generating these codes, and showed how to modify the above algorithm in order to generate all the binary self-dual 1-generator QC codes. Siap and Kulhan [11] have determined the generators of 1-generator GQC codes and proved a BCH type bound for this family of codes over Fq . Recently, Esmaeili and Yari [4] apply the CRT to decompose GQC codes over Fq . They characterize ρ-generator GQC codes in details and give a good lower bound on the minimum distance of such a code in terms of the minimum distance of the constituent codes. When R = Z4 , Pei and Zhang [9] have considered enumeration of QC codes and some specific subclasses of these codes over Z4 . But enumeration of GQC codes and QC codes over arbitrary Galois rings has not been considered to the best of our knowledge. Let p be a prime number, s a positive integer and f (x) a monic basic irreducible polynomial of degree v > 0 over Z ps . It is known that the residue class ring Z ps [x]/( f (x)) is a Galois ring of characteristic p s and cardinality p sv . It is also known that two Galois rings of the same characteristic and cardinality are isomorphic (see McDonald [8] or Wan [13]) The main theme of this present paper is to study GQC codes over Galois rings, focusing on structural properties and enumeration. In Sect. 2, the factorization of polynomials over a Galois ring, a generalization of the CRT to the polynomial ring
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over a Galois ring (Lemma 2.3) and enumeration of submodules of a free module with finite rank over a Galois ring are sketched. In Sect. 3, we investigate structural properties and enumeration of GQC codes over a Galois ring. In Sect. 4, we investigate structural properties and enumeration of 1-generator GQC codes over a Galois ring. In Sect. 5, we investigate structural properties and enumeration of GQC codes with a given parity-check polynomial over a finite field. As corollaries, we obtain structural properties and enumeration of QC codes, respectively. In Sect. 6, we give two algorithms to compute the number of GQC codes and QC codes, respectively. 2 Preliminaries Let R = GR( p s , q s ) be a Galois ring of characteristic p s and cardinality q s , where p is a prime number, q a power of p and s a positive integer. For simplicity, we write a polynomial f (x) as f in the following. Let f 1 , f 2 ∈ R[x]. They are said to be coprime in R[x] if there are λ1 , λ2 ∈ R[x] such that λ1 f 1 + λ2 f 2 = 1, or equivalently R[x] f 1 + R[x] f 2 = R[x]. First, we list results for polynomials in R[x] needed in the following sections. Lemma 2.1 ([12] Theorem 2.4) Let m be a positive integer not divisible by p. Then the polynomial x m − 1 can be factored into a product of some number (say, r) of pairwise coprime monic basic irreducible polynomials f 1 , f 2 , . . . , fr over R. Moreover, f 1 , f 2 , . . . , fr are uniquely determined up to a rearrangement. Lemma 2.2 ([12] Lemma 2.5) Let f 1 , f 2 , . . . , fr be pairwise coprime polynomials in R[x] and fˆi the product of all f j except f i . Then f i and fˆi are coprime in R[x]. Lemma 2.3 ([12] Theorem 2.9) Let f 1 , f 2 , . . . , fr be pairwise coprime monic polynomials of degree ≥ 1 over R, f = f 1 . . . fr and R = R[x]/( f ). By Lemma 2.2, there are bi , ci ∈ R[x] such that bi fˆi + ci f i = 1 in R[x]. Let ei = bi fˆi + ( f ) ∈ R. Then (i) e1 , e2 , . . . , er are mutually orthogonal non-zero idempotents of R. (ii) e1 + · · · + er = 1 in R. (iii) Let Ri = Rei be the principal ideal of R generated by ei . Then ei is the identity of the ring Ri and Ri = Rei = R( fˆi + ( f )). (iv) R = R1 ⊕ · · · ⊕ Rr where ⊕ denotes the direct sum of rings. (v) For each i = 1, . . . , r , the map R[x]/( f i ) → Ri , via g+( f i ) → (g+( f ))ei = gbi fˆi + ( f )(∀ g ∈ R[x]), is a well-defined isomorphism of rings. (vi) R ∼ = R[x]/( f 1 ) ⊕ · · · ⊕ R[x]/( fr ). Lemma 2.4 ([13] Theorem 14.23) Let h be a monic basic irreducible polynomial of degree k over R. Then R[x]/(h) is a Galois ring of characteristic p s and cardinality q sk , i.e. R[x]/(h) = GR( p s , q sk ). Then we list results for R-submodules of R n needed in the following sections. Lemma 2.5 (cf. The right-left dual of [1] Lemma 3.1) Let S be a nonzero R-submodule of R n . Then there exist a unique multiset {s1 , . . . , sk } and a right invertible k × n matrix Q over R such that S is generated by the row vectors of the matrix Q, where 0 ≤ s1 ≤ · · · ≤ sk ≤ s − 1, 1 ≤ k ≤ n and = diag( p s1 , . . . , p sk ).
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If S = 0, in the notation of Lemma 2.5 we call S a (k1 · pr1 , . . . , kt · prt )-type submodule of R n when {s1 , . . . , sk } = {r1k1 , . . . , rtkt } as multisets, where 0 ≤ r1 < · · · < n i −1) (q n n t ki = k. Let = i=n−k+1 and = 1 be rt ≤ s − 1, ki ∈ Z+ and i=1 k (q i −1) i=1 k q 0 q the Gaussian coefficient, where n, k are positive integers. Lemma 2.6 (cf. [1] Lemma 6.1 and Theorem 6.3 and [2] Lemma 2.5) For any 1 ≤ k ≤ n, let {r1k1 , . . . , rtkt } be a multiset, where 0 ≤ r1 < · · · < rt ≤ s − 1, ki ∈ k k t {r1 1 ,...,rt t } Z+ and i=1 ki = k. Let N(n,q,s) be the number of (k1 · pr1 , . . . , kt · prt )-type n R-submodules of R . Then k
k
{r1 1 ,...,rt t } N(n,q,s)
=q
nk(s−1)+ 12 k(k−1)+ 21
t
ki (ki +1)−γ −(n−k)
i=1
t
ki ri
i=1
k1 + · · · + kt k2 + · · · + kt n kt−1 + kt × ··· k1 k2 kt−1 k q q q q
⎞⎞ ⎛ ⎞ 0 r2 − r1 . . . rt − r1 k1 ⎜ ⎜ 0 ⎟⎟ ⎜ k 2 ⎟ 0 . . . r − r t 2 ⎜ ⎟⎟ ⎜ ⎟ and Jt is where γ = (k1 , k2 , . . . , kt ) ⎜ ⎝s Jt − ⎝ . . . ... ... . . . ⎠⎠ ⎝ . . . ⎠ 0 0 ... 0 kt n the all-ones t × t matrix. Especially, the number of free submodules of R with rank k n {0 } . k equals N(n,q,s) = q (n−k)k(s−1) k q ⎛
⎛
Then from Lemma 2.6 we deduce Corollary 2.7 Let N(n,q,s) be the number of R-submodules of R n . Then N(n,q,s) = 1 +
n
k=1 ki ≥1, k1 +···+kt =k 0≤r1
