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

TL;DR

This paper presents algebraic methods to design series of MDS linear block and convolutional codes with specified rate, distance, and type, including infinite series approaching optimal parameters, and applicable to quantum and LCD codes.

Contribution

It introduces algebraic algorithms for designing MDS block and convolutional codes with specific properties and infinite series approaching optimal rate and distance.

Findings

Design of infinite series of block codes with rate approaching R and relative distance approaching (1-R).

Construction of convolutional codes with rate approaching R and distance approaching 2(1-R).

Algebraic properties and decoding methods for these codes are established.

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.