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Modular and p-adic cyclic codes

A. R. Calderbank·N. J. A. Sloane
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Abstract

This paper presents some basic theorems giving the structure of cyclic codes of length n over the ring of integers modulo p^a 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 + lambda X^2 + (lambda - 1) X - 1, where lambda satisfies lambda^2 - lambda + 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-adic Golay code of length 24 and the 3-adic Golay code of length 12.

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A. R. Calderbank, N. J. A. Sloane. 2003-11-18. Modular and p-adic cyclic codes. https://arxiv.org/abs/math/0311319

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