p-Adic Codes

Abstract

The p-adic numbers were first considered by Hensel in the 19th century. He observed that the primes play an analogous role in the integers as linear polynomials do in ${\mathbb C}[X]$. The Laurent expansion of a rational function led him to consider the p-adic expansion of a rational number. In this chapter, for a fixed prime p, we will construct block codes over the rings ${\mathbb Z}/p^h{\mathbb Z}$ simultaneously, by constructing codes over the p-adic numbers and then considering the coordinates modulo p ^h. These codes will be linear over the ring but when mapped to codes over ${\mathbb Z}/p{\mathbb Z}$ will result in codes which are not equivalent to linear codes. We start with a brief introduction to p-adic numbers, which will cover enough background for our purposes. The classical cyclic codes, that we constructed in Chapter 5, lift to cyclic codes over the p-adic numbers. In the case of the cyclic Hamming code, this lift extends to a code over ${\mathbb Z}/4{\mathbb Z}$ which, when mapped to a binary code, gives a non-linear code with a set of parameters for which no linear code exists.