Coding Theory: the unit-derived methodology

TL;DR

The paper introduces the unit-derived methodology in coding theory, demonstrating its optimality for constructing codes with efficient decoding, including MDS codes from Vandermonde or Fourier matrices, and provides explicit constructions and proofs.

Contribution

It presents a novel unit-derived approach that yields optimal codes with practical decoding algorithms, including explicit constructions for Shannon's theorem and finite field optimal codes.

Findings

Unit-derived codes from Vandermonde or Fourier matrices are MDS and have efficient decoding.

Explicit constructions with decoding algorithms are provided for Shannon's theorem.

Codes are constructed to be optimal for specific finite fields.

Abstract

The unit-derived method in coding theory is shown to be a unique optimal scheme for constructing and analysing codes. In many cases efficient and practical decoding methods are produced. Codes with efficient decoding algorithms at maximal distances possible are derived from unit schemes. In particular unit-derived codes from Vandermonde or Fourier matrices are particularly commendable giving rise to mds codes of varying rates with practical and efficient decoding algorithms. For a given rate and given error correction capability, explicit codes with efficient error correcting algorithms are designed to these specifications. An explicit constructive proof with an efficient decoding algorithm is given for Shannon's theorem. For a given finite field, codes are constructed which are `optimal' for this field.