Cyclic codes over

2012

The ring in the title is the first non commutative ring to have been used as alphabet for block codes. The original motivation was the construction of some quaternionic modular lattices from codes. The new application is the construction of space time codes obtained by concatenation from the Golden code. In this article, we derive structure theorems for cyclic codes over that ring, and use them to characterize the lengths where self dual cyclic codes exist. These codes in turn give rise to formally self dual quaternary codes.

Journal of the Franklin Institute, 2013

The ring in the title is the first non commutative ring to have been used as alphabet for block codes. The original motivation was the construction of some quaternionic modular lattices from codes. The new application is the construction of space time codes obtained by concatenation from the Golden code. In this article, we derive structure theorems for cyclic codes over that ring, and use them to characterize the lengths where self dual cyclic codes exist. These codes in turn give rise to formally self dual quaternary codes.

Bulletin of the Korean Mathematical Society, 2016

In this paper, we extend the results given in [3] to a nonchain ring Rp = Fp + vFp + • • • + v p−1 Fp, where v p = v and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring Rp and study the structure of their duals. We classify cyclic codes containing their duals over Rp by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map π defined in [4], we give the parameters of the quantum codes of length pn over Fp which are obtained from cyclic codes over Rp. Finally, we illustrate the results by giving some examples.

2016

We propose a full-rate iterated space-time code construction, to design codes of Q-rank 2n from cyclic algebra based codes of Q-rank n. We give a condition for determining whether the resulting codes satisfy the full diversity property, and study their maximum likelihood decoding complexity with respect to sphere decoding. Particular emphasis is given to the asymmetric MIDO (multiple input double output) codes. In the process, we derive an interesting way of obtaining division algebras, and study their center and maximal subfield.

2015 6th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO), 2015

In this paper, we extend the results given in [3] to a nonchain ring Rp = Fp + vFp + • • • + v p−1 Fp, where v p = v and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring Rp and study the structure of their duals. We classify cyclic codes containing their duals over Rp by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map π defined in [4], we give the parameters of the quantum codes of length pn over Fp which are obtained from cyclic codes over Rp. Finally, we illustrate the results by giving some examples.

arXiv (Cornell University), 2012

We propose a full-rate iterated space-time code construction, to design codes of Q-rank 2n from cyclic algebra based codes of Q-rank n. We give a condition for determining whether the resulting codes satisfy the full diversity property, and study their maximum likelihood decoding complexity with respect to sphere decoding. Particular emphasis is given to the asymmetric MIDO (multiple input double output) codes. In the process, we derive an interesting way of obtaining division algebras, and study their center and maximal subfield.

Discrete Mathematics, 1997

It is well known that cyclic linear codes of length n over a (finite) field F can be characterized in terms of the factors of the polynomial x"-1 in F[x]. This paper investigates cyclic linear codes over arbitrary (not necessarily commutative) finite tings and proves the above characterization to be true for a large class of such codes over these rings. (~

In this study, we consider linear and especially cyclic codes over the non-chain ring $Z_p[v]/<v^p − v>$ where $p$ is a prime. This is a generalization of the case $p = 3.$ Further, in this work the structure of constacyclic codes are studied as well. This study takes advantage mainly from a Gray map which preserves the distance between codes over this ring and $p-$ary codes and moreover this map enlightens the structure of these codes. Furthermore, a MacWilliams type identity is presented together with some illustrative examples.

Information Theory, IEEE Transactions on, 2001

A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese Remainder Theorem (CRT), or of the Discrete Fourier Transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the recent ( + ) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. Other results made possible by the ring decomposition are a characterization of self-dual quasi-cyclic codes, and a trace representation that generalizes that of cyclic codes.

IEEE Transactions on Information Theory, 2002

Recently, (linear) codes over and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to produces a new binary code, a (92 2 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials.