On the additive cyclic structure of quasi-cyclic codes

An index $?$, length $m?$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field ${\mathbb{F}}_{q^{?}}$ via a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. However, this cyclic code is only linear over ${\mathbb{F}}_q$, making it an additive cyclic code, or an ${\mathbb{F}}_q$-linear cyclic code, over the alphabet ${\mathbb{F}}_{q^{?}}$. This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have ${\mathbb{F}}_{q^{?}}$-linear cyclic images under a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes.     

On the Hamming distances of repeated-root constacyclic codes of length 4ps

Let $p$ be an odd prime, $s$, $m$ be positive integers, $\gamma ,\lambda$ be nonzero elements of the finite field ${\mathbb{F}}_{p^m}$ such that ${\gamma }^{p^s}=\lambda$. In this paper, we show that, for any positive integer $\eta$, the Hamming distances of all repeated-root $\lambda$-constacyclic codes of length $\eta p^s$ can be determined by those of certain simple-root $\gamma$-constacyclic codes of length $\eta$. Using this result, Hamming distances of all constacyclic codes of length $4p^s$ are obtained. As an application, we identify all MDS $\lambda$-constacyclic codes of length $4p^s$.     

泛逻辑的零级泛运算模型的代数性质

讨论泛逻辑的零级泛运算模型的基本代数性质。证明T（x,y,^）是阿基米德型三角范数;泛与运算模型与泛蕴涵运算模型形成一个伴随对;当h∈（0,0．75）时,有界格（[0,1],∨,∧,,＊,→0,1）做成一个MV-代数;当h∈（0．75,1）时,有界格（[0,1],∨,∧,＊,→0,1）做成一个乘积代数。进一步,给出了零级泛与运算模型与泛或运算模型的加性生成元与乘性生成元。     

Characterizations of some two-dimensional cyclic codes correspond to the ideals of F[x,y]/

Two-dimensional cyclic code is one of the natural generalizations of cyclic code. In this paper we study the algebraic structure of some two-dimensional cyclic codes and their dual codes.     

A note on the representation for the Drazin inverse of block matrices

In this note we consider representations of the Drazin inverse of 2×2 block matrices under conditions weaker than those used in recent papers on the subject, in particular in [D.S. Djordjević, P.S. Stanimirović, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (126) (2001) 617–634; R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2×2 block matrix, SIAM J. Matrix Anal. Appl. 27 (2006) 757–771].     

Orbit matrices of Hadamard matrices and related codes

In this paper we introduce the notion of orbit matrices of Hadamard matrices with respect to their permutation automorphism groups and show that under certain conditions these orbit matrices yield self-orthogonal codes. As a case study, we construct codes from orbit matrices of some Paley type I and Paley type II Hadamard matrices. In addition, we construct four new symmetric (100,45,20) designs which correspond to regular Hadamard matrices, and construct codes from their orbit matrices. The codes constructed include optimal, near-optimal self-orthogonal and self-dual codes, over finite fields and over ${\mathbb{Z}}_4$.     

Extremal Ternary Self-Dual Codes Constructed from Negacirculant Matrices

In this paper, a construction of ternary self-dual codes based on negacirculant matrices is given. As an application, we construct new extremal ternary self-dual codes of lengths 32, 40, 44, 52 and 56. Our approach regenerates all the known extremal self-dual codes of lengths 36, 48, 52 and 64. New extremal ternary quasi-twisted self-dual codes are also constructed. Supported by an NSERC discovery grant and a RTI grant. Supported by an NSERC discovery grant and a RTI grant. A summer student Chinook Scholarship is greatly appreciated.     

LDPC codes from the Hermitian curve

In this paper, we study the code which has as parity check matrix the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in (Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that has a double cyclic structure and that by shortening in a suitable way it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix via a geometric approach.      

环F2+uF2上线性码的二元像

在有限环R=F2+uF2与F2之间定义了一个新的Gray映射,给出了环F2+uF2上线性码C的二元像φ(C)的生成矩阵,证明了环F2+uF2上线性码C及其对偶码的二元像仍是对偶码.     

Generalized weighing matrices and self-orthogonal codes

It is shown that the row spaces of certain generalized weighing matrices, Bhaskar Rao designs, and generalized Hadamard matrices over a finite field of order q are hermitian self-orthogonal codes. Certain matrices of minimum rank yield optimal codes. In the special case when q=4, the codes are linked to quantum error-correcting codes, including some codes with optimal parameters.