Sequences Which Are Period Invariant Under Rearrangement Of The Elements: Relations To Generalized Hadamard Codes

摘要

Rings of polynomials R_N = Z_p[x]/x~N -1 whichrnare isomorphic to Z_p~N are studied, where p is prime and N is an integer. If I is an ideal in R_N, the code K whose vectors constitute thernisomorphic image of I is a linear cyclic code. If I is a principle ideal and K contains only the trivial cycle {0} and one nontrivial cycle of maximal least period N, then the code words of K/{0} obtained by removing the zero vector can be arranged in an order which constitutes a linear circulant matrix, C. The distribution of the elements of C is such that it forms the cyclic core of a generalized Hadamard matrix ever the additive group of Z_p. A necessary conditionrnthat C=K/{0} be linear circulant is that for each row vector v of C, the periodic infinite sequence a(v) produced by cycling the elements of v be period invariant under an arbitrary permutation of the elements of the first period. The necessary and sufficient condition that C be linear circulant is that the dual ideal generated by the parity check polynomial h(x) of K be maximal (a non-trivial, prime ideal of R_N),rnwith N = p~k -1 and k - deg( h(x)).