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

Maximum Distance Separable Codes to Order Even better though is GF (53) = Z 53 which is a prime field. This has an element of order 52 from which the Fourier 52 × 52 matrix can be formed. Now ω = (2 mod 53) is an element of order 52 in GF (53). Work and codes with the resulting Fourier 52 × 52 matrix can then be done in modular arithmetic, within Z 53 , using powers of (2 mod 53). This section is for information on developments and is not required subsequently. Particular types of MDS codes may be required. These are not dealt with here but the following is noted. • A quantum MDS code is one of the form [[ n, r, d ]] where 2 d = n − r + 2, see [20] for details. In [12] the methods are applied to construct and develop MDS quantum codes of different types and to required specifications. This is done by requiring the constructed codes to be dual-containing MDS codes from which quantum MDS error-correcting codes are constructed from the CSS construction developed in [2, 3]. This is further developed for the construction and development of Entanglement assisted quantum error-correcting codes , EAQECC, of different types and to required specifications in [8]. • In [10] Linear complementary dual (LCD), MDS codes are constructed based on the general con- structions. An LCD code C is a code such that C ∩ C ⊥ = 0. These have found use in security, in data storage and communications’ systems. In [10] the rows are chosen according to a particular formulation so as to derive LCD codes which are also MDS codes. • In [15] error-correcting codes, similar to ones here, are used for solving underdetermined systems of equations for use in compressed sensing . • By using rows of the Fourier matrix as matrices for polynomials, MDS convolutional codes, achiev- ing the generalized Singleton bound see [21], are constructed and analysed in [14]. • The codes developed here seem particularly suitable for use in McEliece type encryption/decryption, [19]; this has yet to be investigated. 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 1 ( F ) © 2021 Global Journals Version I i. Developments on different types of MDS codes that can be constructed R eferences R éférences R eferencias 1. Richard E. Blahut, Algebraic Codes for data transmission, Cambridge University Press, 2003. 2. A.R. Calderbank, E.M. Rains, P.M. Shor, N.J.A. Sloane, “ Quantum error correction via codes over GF(4) ” , IEEE Trans. on Information Theory, 44(4), 1369-1387, 1998. 3. Calderbank, A.R. and Shor, P.W, “ Good quantum error-correcting codes exist. ” Phys. Rev. A, 54, no. 2, 1098-1105, 1996. 4. Keith Conrad ’ s notes on ‘ Cyclicity of (Z/(p)) × ’ , available on the internet. 5. GAP – Groups, Algorithms, and Programming, (https://www.gap-system.org ) 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. 7. Barry Hurley, Ted Hurley, “ Systems of mds codes from units and idempotents ” , Discrete Math. 335, 81-91, 2014. 8. Ted Hurley, Donny Hurley, Barry Hurley, “ Entanglement-assisted quantum error- correcting codes from units ” , arXiv:1806.10875. 9. Paul Hurley and Ted Hurley, “ Codes from zero-divisors and units in group rings ” ; Int. J. of Information and Coding Theory, Vol. 1, 1, 57-87, 2009. 8. Ted Hurley, Donny Hurley, Barry Hurley, “ Entanglement-assisted quantum error- correcting codes from units ” , arXiv:1806.10875. R ef