Introduction

Let K be a finite field and let V be the vector space of n-tuples over K. The Hamming-distance between elements x and y of V, denoted d(x, y), is defined by d(x, y) := #{ 1 <= i <= n | x_i != y_i }. The minimum distance d for a subset C of V is then d = min{ d(x, y) | x in C, y in C, x != y }. The subset C of V is called an (n, M, d) code if the minimum distance for the subset C is d and |C| = M. The code C is called a [n, k, d] linear code if C is a k-dimensional subspace of V. Then V is also called the ambient space of C.

Currently, Magma supports only linear codes. In this chapter, the term "code" will refer to a linear code. Magma provides machinery for studying linear codes over finite fields F_q=GF(q) and over the finite rings Z_m=Z/mZ (the integer residue classes modulo m).

This chapter describes those functions which are applicable to codes over F_q. The highlights of the facilities provided for such codes include:

• The construction of codes in terms of generator matrices, parity check matrices and generating polynomials (cyclic codes).

• A large number of constructions for particular families of codes, e.g., quadratic residue codes.

• Highly optimized algorithms for the calculation of the minimum weight.

• Various forms of weight enumerator including the Macwilliams transform.

• A database that gives the user access to every best known linear code over GF(2) of length up to 256.

• Machinery that allows the user to construct algebraic--geometric codes from a curve defined over F_q.

• The computation of automorphism groups for codes over small fields.