Title: Coding theory: the unit-derived methodology

Authors: Ted Hurley; Donny Hurley

Addresses: Department of Mathematics, National Universiy of Ireland Galway, Galway, Ireland ' Department of Computing and Electronic Engineering, Institute of Technology Sligo, Ireland

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.

Keywords: code; unit-derived schemes; decoding.

DOI: 10.1504/IJICOT.2018.091852

International Journal of Information and Coding Theory, 2018 Vol.5 No.1, pp.55 - 80

Received: 08 Feb 2018
Accepted: 10 Feb 2018
Published online: 18 May 2018 *