Cyclic and BCH codes whose minimum distance equals their maximum BCH bound (original) (raw)

A BCH code and a sequence of cyclic codes

International Journal of Algebra, 2014

This study establishes that for a given binary BCH code C 0 n of length n generated by a polynomial g(x) ∈ F 2 [x] of degree r there exists a family of binary cyclic codes {C m 2 m−1 (n+1)n } m≥1 such that for each m ≥ 1, the binary cyclic code C m 2 m−1 (n+1)n has length 2 m−1 (n + 1)n and is generated by a generalized polynomial g(x 1 2 m) ∈ F 2 [x, 1 2 m Z ≥0 ] of degree 2 m r. Furthermore, C 0 n is embedded in C m 2 m−1 (n+1)n and C m 2 m−1 (n+1)n is embedded in C m+1 2 m (n+1)n for each m ≥ 1. By a newly proposed algorithm, codewords of the binary BCH code C 0 n can be transmitted with high code rate and decoded by the decoder of any member of the family {C m 2 m−1 (n+1)n } m≥1 of binary cyclic codes, having the same code rate.

From ds-bounds for cyclic codes to true distance for abelian codes

ArXiv, 2017

In this paper we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes through the notion of mathbbB\mathbb{B}mathbbB-apparent distance. We use this technique to improve the searching for new bounds for the minimum distance of abelian codes. We also study conditions for an abelian code to verify that its mathbbB\mathbb{B}mathbbB-apparent distance reaches its (true) minimum distance. Then we construct some tables of such codes as an application

A New Bound on the Minimum Distance of Cyclic Codes Using Small-Minimum-Distance Cyclic Codes

2012

A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann-Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes.

From ds-Bounds for Cyclic Codes to True Minimum Distance for Abelian Codes

IEEE Transactions on Information Theory, 2019

In this paper we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes through the notion of B-apparent distance. We also study conditions for an abelian code to verify that its B-apparent distance reaches its (true) minimum distance. Then we construct some codes as an application.

On the Minimal Distance of Binary Self-Dual Cyclic Codes

IEEE Transactions on Information Theory, 2000

In this paper, an explicit construction of binary selfdual cyclic codes of length n going to infinity with a minimal distance at least half the square root of n is presented. The same idea is also used to construct more general binary cyclic codes with a large minimal distance. Finally, in the special case of self-dual cyclic codes, a simplified version of a proof by Conway and Sloane is given, showing an upper bound for the distance of binary selfdual codes.

Generalizing Bounds on the Minimum Distance of Cyclic Codes Using Cyclic Product Codes

2013

Two generalizations of the Hartmann-Tzeng (HT) bound on the minimum distance of q-ary cyclic codes are proposed. The first one is proven by embedding the given cyclic code into a cyclic product code. Furthermore, we show that unique decoding up to this bound is always possible and outline a quadratic-time syndrome-based error decoding algorithm. The second bound is stronger and the proof is more involved.

Efficient decoding of some classes of binary cyclic codes beyond the Hartmann-Tzeng bound

2011

A new bound on the distance of binary cyclic codes is proposed. The approach is based on the representation of a subset of the roots of the generator polynomial by a rational function. A new bound on the minimum distance is proven and several classes of binary cyclic codes are identified. For some classes of codes, this bound is better than the known bounds (e.g. BCH or Hartmann-Tzeng bound). Furthermore, a quadratic-time decoding algorithm up to this new bound is developed.

CONSTRUCTION OF QUASI-CYCLIC CODES

1989

The class of Quasi-Cyclic Error Correcting Codes is investigated. It is shown that they contain many of the best known binary and nonbinary codes. Tables of rate 1/p and (p − 1)/p Quasi-Cyclic (QC) codes are constructed, which are a compilation of previously best known codes as well as many new codes constructed using exhaustive, and other more sophisticated search techniques. Many of these binary codes attain the known bounds on the maximum possible minimum distance, and 13 improve the bounds. The minimum distances and generator polynomials of all known best codes are given. The search methods are outlined and the weight divisibility of the codes is noted. The weight distributions of some s-th Power Residue (PR) codes and related rate 1/s QC codes are found using the link established between PR codes and QC codes. Subcodes of the PR codes are found by deleting certain circulant matrices in the corresponding QC code. They are used as a starting set of circulants for other techniques. Nonbinary Power Residue codes and related QC codes are constructed over GF(3), GF(4), GF(5), GF(7) and GF(8). Their subcodes are also used to find good nonbinary QC codes. A simple and efficient algorithm for constructing primitive polynomials with linearly independent roots over the Galois Field of q elements, GF(q), is developed. Tables of these polynomials are presented. These Tables are unknown for polynomials with nonbinary coefficients, and the known binary Tables are incomplete. The polynomials are employed in such diverse areas as construction of error correcting codes, efficient VLSI implementation of multiplication and inverse operations over Galois Fields, and digital testing of integrated circuits. Using the link established between generalized tail biting convolutional codes and binary QC codes, good QC codes are constructed based on iii Optimum Distance Profile (ODP) convolutional codes. Several best rate 2/3 systematic codes up to circulant size 20 are constructed in this manner.