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 ∈C, y ∈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. Then V is referred to as the ambient space of C.

The code C is called a [n, k, d] linear code if C is a k-dimensional subspace of V. Currently Magma supports not only linear codes, but also codes over finite fields which are only linear over some subfield. These are known as additive codes and can be found in Chapter ADDITIVE CODES. This chapter deals only with 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), over the integer residue classes Z_m = Z/mZ, and over galois rings GR(pⁿ, k).

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 best known linear codes GF(q) for q=2, 3, 4, 5, 7, 8, 9.

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.