Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 21 Issue 4

Maximum Distance Separable Codes to Order Ted Hurley α , Donny Hurley σ & Barry Hurley ρ I. I ntroduction 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 1 ( F ) © 2021 Global Journals Version I Coding theory is at the heart of modern day communications. Maximum distance separable, MDS, codes are at the heart of coding theory. Data needs to be transmitted safely and sometimes securely. Best rate and error-correcting capabilities are the aim, and MDS codes can meet the requirements; they correct the maximum number of errors for given length and dimension. General methods for constructing MDS codes over finite fields are given in Section 2 following [6, 7, 15]. The codes are explicitly constructed over finite fields with efficient encoding and decoding algorithms of complexity max { O ( n log n ) , O ( t 2 ) } , where t is the error-correcting capability. These are exploited. For given { n, r } MDS ( n, r ) codes are constructed over finite fields with characteristics not dividing n , section 3.1. For given rate and given error-correcting capability series of MDS codes to these specifications are constructed over finite fields, section 3.2. For given rate R , 0 < R < 1, series of MDS codes are constructed over finite fields in which the ratio of the distance by the length approaches (1 − R ), section 3.3. For a given finite field GF ( q ), MDS ( q − 1 , r ) codes of different types are constructed over GF ( q ) for any given r, 1 ≤ r ≤ ( q − 1), section 3.7. The codes are explicit with efficient encoding and decoding algorithms as noted. In addition for each n/ ( q − 1), MDS codes of length n and dimension r are constructed over GF ( q ) for any given r, 1 ≤ r ≤ n . In particular for p a prime, MDS ( p − 1 , r ) codes are constructed in GF ( p ) = Z p in which case the arithmetic is modular arithmetic which works smoothly and very efficiently. For given R = r n , 0 < R < 1, with p 6 | n , series of codes over finite fields of characteristic p are constructed in which the ratio of the distance to the length approaches (1 − R ), section 3.4. Note 0 < R < 1 if and only if 0 < (1 − R ) < 1. In particular such series are constructed in fields of characteristic 2 for cases where the denominator n of the given rate is odd. Author α : National University of Ireland Galway. e-mail: Ted.Hurley@NuiGalway.ie Author σ : Institute of Technology, Sligo. e-mail: hurley.donny@itsligo.ie Author ρ : e-mail: barryj 2000@yahoo.co.uk Abstract- Maximum distance separable (MDS) are constructed to required specifications. The codes are explicitly given over finite fields with efficient encoding and decoding algorithms. Series of such codes over finite fields with ratio of distance to length approaching (1 − R ) for given R , 0 < R < 1 are derived. For given rate R = , with p not dividing n, series of codes over finite fields of characteristic p are constructed such that the ratio of the distance to the length approaches (1 − R ). For a given field GF(q) MDS codes of the form ( q −1, r ) are constructed for any r . The codes are encompassing, easy to construct with efficient encoding and decoding algorithms of complexity max { O ( n log n ), t 2 }, where t is the error-correcting capability of the code. r n 6. Ted Hurley and Donny Hurley, “ Coding theory: the unit-derived methodology ” , Int. J. Information and Coding Theory, Vol. 5, no.1, 55-80, 2018. R ef