Convolutional codes from unit schemes

Abstract

Convolutional codes are constructed, designed and analysed using row and/or block structures of unit algebraic schemes. Infinite series of such codes and of codes with specific properties are derived. Properties are shown algebraically and algebraic decoding methods are derived. For a given rate and given error-correction capability at each component, convolutional codes with these specifications and with efficient decoding algorithms are constructed. Explicit prototype examples are given but in general large lengths and large error capability are achievable. Convolutional codes with efficient decoding algorithms at or near the maximum free distances attainable for the parameters are constructible. Unit memory convolutional codes of maximum possible free distance are designed with practical algebraic decoding algorithms. LDPC (low density parity check) convolutional codes with efficient decoding schemes are constructed and analysed by the methods. Self-dual and dual-containing convolutional codes may also be designed by the methods; dual-containing codes enables the construction of quantum codes.