Modular and p-adic Cyclic Codes

This paper presents some basic theorems giving the structure of cyclic codes of length n over the ring of integers modulo pa and over the p-adic numbers, where p is a prime not dividing n. An especially interesting example is the 2-adic cyclic code of length 7 with generator polynomial X 3 + ,~X 2 + (L I)X -l, where )~ satisfies ~2 _ k + 2 = 0. This is the 2-adic generalization of both the binary Hamming code and the quaternary octacode (the latter being equivalent to the Nordstrom-Robinson code). Other examples include the 2-adie Golay code of length 24 and the 3-adic Golay code of length 12.