ON THE GENERATORS OF CODES OF IDEALS OF THE POLYNOMIAL RING FOR ERROR CONTROL

Abstract

Shannon introduced error detection and correction codes to address the growing need of efficiency and reliability of code vectors. Ideals in algebraic number system have mainly been used to preserve the notion of unique factorization in rings of algebraic integers and to prove Fermat's Last Theorem. Generators of codes of ideals of polynomial rings have not been fully characterized. Ideals in Noetherian rings are closed in polynomial addition and multiplication. This property has been used to characterize cyclic codes. This class of cyclic codes has a rich algebraic structure which is a valuable tool in coding design. The Golay Field which has been used to generate codes over the years provides codes of fixed length which do not reach Shannon's limit. This research has used Shannon's proposed model to determine generators of codes of ideals of the polynomial ring to be used for error control. It presents generators of codes of ideals of the polynomial ring associated with the codewords of a cyclic code C. If the set of generator polynomials corresponding to codewords is given by I(C) (a set of principal ideals of the polynomial ring), it has been shown that I(C) is a cyclic code. Additionally the suitability of codes of ideals of the polynomial ring for error control has been established. Application of Shannon's Theorem on optimal codes has been done to characterize generators of codes of ideals of the polynomial ring for error control. The generators of codes of the candidate polynomial ring Fn2 [x]/hxn􀀀1i have been investigated and characterized using lattices, simplex Hamming codes and isometries. The results of this research contribute significantly towards characterization of generators of codes from ideals of polynomial rings.