Linear block and convolutional MDS codes to required rate, distance and type

Abstract

Algebraic methods for the design of series of maximum distance separable (MDS) linear block and convolutional codes to required specifications and types are presented. Algorithms are given to design codes to required rate and required error-correcting capability and required types. Infinite series of block codes with rate approaching a given rational $R$ with $0<R<1$ and relative distance over length approaching $(1-R)$ are designed. These can be designed over fields of given characteristic $p$ or over fields of prime order and can be specified to be of a particular type such as (i) dual-containing under Euclidean inner product, (ii) dual-containing under Hermitian inner product, (iii) quantum error-correcting, (iv) linear complementary dual (LCD). Convolutional codes to required rate and distance and infinite series of convolutional codes with rate approaching a given rational $R$ and distance over length approaching $2(1-R)$ are designed. The designs are algebraic and properties, including distances, are shown algebraically. Algebraic explicit efficient decoding methods are referenced.

References in corpus (11)

• LCD codes over ${\mathbb{F}}_q$ are as good as linear codes for q at least four
• Quantum Convolutional BCH Codes
• Quantum Block and Convolutional Codes from Self-orthogonal Product Codes
• Simple proof of Chebotarev's theorem on roots of unity
• Convolutional codes from units in matrix and group rings
• Self-dual, dual-containing and related quantum codes from group rings
• New characterization and parametrization of LCD Codes
• The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices
• Linear complementary dual, maximum distance separable codes
• Maximum distance separable codes to order
• Quantum error-correcting codes: the unit-derived strategy

Cited by in corpus (1)

• Ultimate linear block and convolutional codes