Algebraically Punctured Cyclic Codes

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 (qk − 1, k) code. The algorithm for puncturing is algebraic in that the coordinates deleted form subgroups of the additive group of the (qk − 1) roots of unity, or cosets of the multiplicative group of the (qk − 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 qk−1. Optimality is proven by generalizing the Griesmer Bound on group codes. Encoding and decoding procedures are presented for this class of codes.