Algebraically punctured cyclic codes

Abstract

We present a new class of optimal ( n, k ) group codes over the general finito field GF ( q ), q , a prime power, which are obtained by systematically deleting or puncturing certain coordinates of the maximal length shift register ( q k − 1, k ) code. The algorithm for puncturing is algebraic in that the coordinates deleted form subgroups of the additive group of the ( q k − 1) roots of unity, or cosets of the multiplicative group of the ( q k − 1) roots of unity, modulo the multiplicative group of GF ( q ). The specific algebraic nature of this puncturing procedure for any particular k yields codes of length n greater than q k−1 . Optimality is proven by generalizing the Griesmer Bound on group codes. Encoding and decoding procedures are presented for this class of codes.