Rank weight hierarchy of some classes of cyclic codes

We study the rank weight hierarchy, thus in particular the rank metric, of cyclic codes over the finite field $\mathbb F_{q^m}$, $q$ a prime power, $m \geq 2$. We establish the rank weight hierarchy for $[n,n-1]$ cyclic codes and characterize $[n,k]$ cyclic codes of rank metric 1 when (1) $k=1$, (2) $n$ and $q$ are coprime, and (3) the characteristic $char(\mathbb F_q)$ divides $n$. Finally, for $n$ and $q$ coprime, cyclic codes of minimal $r$-rank are characterized, and a refinement of the Singleton bound for the rank weight is derived

Weight hierarchies of a family of linear codes associated with degenerate quadratic forms

We restrict a degenerate quadratic form $f$ over a finite field of odd characteristic to subspaces. Thus, a quotient space related to $f$ is introduced. Then we get a non-degenerate quadratic form induced by $f$ over the quotient space. Some related results on the subspaces and quotient space are obtained. Based on this, we solve the weight hierarchies of a family of linear codes related to $f.$Comment: 12 page

Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes

We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on Information Theor

A Class of Three-Weight Cyclic Codes

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a class of three-weight cyclic codes over \gf(p) whose duals have two zeros is presented, where $p$ is an odd prime. The weight distribution of this class of cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a subclass of the cyclic codes are also studied and proved to be optimal.Comment: 11 Page

Rank equivalent and rank degenerate skew cyclic codes

Two skew cyclic codes can be equivalent for the Hamming metric only if they have the same length, and only the zero code is degenerate. The situation is completely different for the rank metric, where lengths of codes correspond to the number of outgoing links from the source when applying the code on a network. We study rank equivalences between skew cyclic codes of different lengths and, with the aim of finding the skew cyclic code of smallest length that is rank equivalent to a given one, we define different types of length for a given skew cyclic code, relate them and compute them in most cases. We give different characterizations of rank degenerate skew cyclic codes using conventional polynomials and linearized polynomials. Some known results on the rank weight hierarchy of cyclic codes for some lengths are obtained as particular cases and extended to all lengths and to all skew cyclic codes. Finally, we prove that the smallest length of a linear code that is rank equivalent to a given skew cyclic code can be attained by a pseudo-skew cyclic code. Throughout the paper, we find new relations between linear skew cyclic codes and their Galois closures