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Let p be an odd prime and m and s positive integers, with m even. Let further F p m be the finite field of p m elements and R = F p m + u F p m ( u 2 = 0 ). Then R is a finite chain ring of p 2 m elements, and there is a Gray map from R N onto F p m 2 N which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dual codes of length 2N over F p m by constructing self-dual codes of length N over R. In particular, it has been shown that one of the key problems in constructing self-dual repeated-root cyclic codes over R is to find an effective way to present precisely Hermitian self-dual cyclic codes of length p s over R. But so far, only the number of these codes has been determined in literature. In this paper, we give an efficient way of constructing all distinct Hermitian self-dual cyclic codes of length p s over R by using column vectors of Kronecker products of matrices with specific types. Furthermore, we provide an explicit expression to present precisely all these Hermitian self-dual cyclic codes, using binomial coefficients. [ABSTRACT FROM AUTHOR]

Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let C be an [n,k] linear code over F{double-struck}q with parity-check matrix H, where the rows of H may be dependent. Let [n] = {1,2,⋯,n} denote the set of column indices of H. A stopping set S of C with parity-check matrix H is a subset of [n] such that the restriction of H to S does not contain a row of weight 1. The stopping set distribution {T i(H)}ni=0 enumerates the number of stopping sets with size i of C with parity-check matrix H. Denote H*, the parity-check matrix, consisting of all the nonzero codewords in the dual code C⊥. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with parity-check matrix H*. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed-Solomon codes, it is easy to determine all the stopping sets. Then, we consider the AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then, the stopping sets, the stopping set distribution, and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting, and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of the AG codes from elliptic curves. © 2014 IEEE.

Let p be any odd prime number and let m, k be arbitrary positive integers. The construction for self-dual cyclic codes of length p k over the Galois ring GR (p 2 , m) is the key to construct self-dual cyclic codes of length p k n over the integer residue class ring Z p 2 for any positive integer n satisfying gcd (p , n) = 1 . So far, existing literature has only determined the number of these self-dual cyclic codes (Des Codes Cryptogr 63:105–112, 2012). In this paper, we give an efficient construction for all distinct self-dual cyclic codes of length p k over GR (p 2 , m) by using column vectors of Kronecker products of matrices with specific types. On this basis, we further obtain an explicit expression for all these self-dual cyclic codes by using binomial coefficients. [ABSTRACT FROM AUTHOR]

We correct some mistakes in the paper "A mass formula for negacyclic codes of length 2k and some good negacyclic codes over ℤ 4 + u ℤ 4 " (Bandi et al. Cryptogr. Commun. 9, 241–272, 2017). [ABSTRACT FROM AUTHOR]

Letbe a finite field of cardinalityqwhereqis a power of an odd prime integer, and denote the generalized quaternion groupby the presentation:wherenis even and satisfies. Left ideals of the group algebraare called left quaternion codes overof length, and abbreviated as left-codes. In this paper, a system theory for left-codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left-code is a direct sum of concatenated codes with the inner codeand the outer code, whereis a minimal self-reciprocal cyclic code overof lengthnandis a skew constacyclic code of length 2 over an extension field or an extension principal ideal ring of. Then we give explicit expressions for outer codes in the concatenated codes, and present the dual code for any left-code precisely. Moreover, all distinct left-codes overand all distinct left-codes overare presented, respectively. [ABSTRACT FROM PUBLISHER]

Let $${\mathbb {F}}_{2^m}$$ be a finite field of characteristic 2 and $$R={\mathbb {F}}_{2^m}[u]/\langle u^k\rangle ={\mathbb {F}}_{2^m} +u{\mathbb {F}}_{2^m}+\ldots +u^{k-1}{\mathbb {F}}_{2^m}$$ ( $$u^k=0$$ ) where $$k\in {\mathbb {Z}}^{+}$$ satisfies $$k\ge 2$$ . For any odd positive integer n, it is known that cyclic codes over R of length 2 n are identified with ideals of the ring $$R[x]/\langle x^{2n}-1\rangle $$ . In this paper, an explicit representation for each cyclic code over R of length 2 n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2 n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2 n are investigated. [ABSTRACT FROM AUTHOR]

Let $$\mathbb {F}_q$$ be a finite field of cardinality $$q$$ , $$l$$ a prime number and $$\mathbb {F}_{q^l}$$ an extension field of $$\mathbb {F}_q$$ with degree $$l$$ . The structure and canonical form decompositions of semisimple multivariable $$\mathbb {F}_q$$ -linear codes over $$\mathbb {F}_{q^l}$$ are presented. Enumeration and construction of these codes are then investigated. Especially, dual codes, self-orthogonality and self-duality of semisimple abelian $$\mathbb {F}_q$$ -linear codes over $$\mathbb {F}_{q^l}$$ are studied. Furthermore, self-dual and $$\gamma $$ -self dual semisimple abelian $$\mathbb {F}_q$$ -linear codes over $$\mathbb {F}_{q^2}$$ are considered. [ABSTRACT FROM AUTHOR]

Let R = Z 4 be the integer ring mod 4. A double cyclic code of length ( r , s ) over R is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . In this paper, we determine the generator polynomials of this family of codes as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Further, we also give the minimal generating sets of this family of codes as R -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual. [ABSTRACT FROM AUTHOR]