Abstract
The study of ideals in algebraic number system has contributed immensely in preserving the notion of unique factorization in rings of algebraic integers and in proving Fermat's last Theorem. Recent research has revealed that ideals in Noethe-rian rings are closed in polynomial addition and multiplication.This property has been used to characterize the polynomial ring Fn 2 [x] mod (xn 1) for error control. In this research we generate ideals of the polynomial ring using GAP software and characterize the polycodewords using Shannon's Code region and Manin's bound.
Highlights
The modern approach to error control coding in digital communication systems was started by Shannon [1], Golay [2] and Hamming [3]
According to Wicker [4] this Hamming code had some undesirable properties; first, it was not efficient, requiring three check bits for every four data bits and second, it could only correct a single error within the block
Golay code [2] addressed these shortcomings by generalizing the construction of the Hamming code
Summary
1.1 Error control CodingThe modern approach to error control coding in digital communication systems was started by Shannon [1], Golay [2] and Hamming [3]. Key Words and Phrases: Polynomial ring; Error detection; Error correction; Code region. According to Wicker [4] this Hamming code had some undesirable properties; first, it was not efficient, requiring three check bits for every four data bits and second, it could only correct a single error within the block.
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