Abstract :
Rings of polynomials RN = Zp[x]/xN − 1 which are isomorphic to ZPN are studied, where p is prime and N is an integer. If I is an ideal in RN, the code K whose vectors constitute` the isomorphic 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 over the additive group of ZPp. A necessary condition that 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(χ) of K be maximal (a nontrivial, prime ideal of RN), with N = pk − 1 and k = deg (h(χ)).
