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

eciency 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 xed

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 Fn

2 [x]/hxn􀀀1i have been investigated and characterized using lattices, simplex Hamming

codes and isometries. The results of this research contribute signicantly towards

characterization of generators of codes from ideals of polynomial rings