## SELF-DUAL CODES

This page is supposed to be a comprehensive database of self-dual codes with the highest known minimum distance. Therefore this place should be in constant change and any questions, suggestions or precisions are welcome.

1. INTRODUCTION
2. GENERAL CONSTRUCTIONS
3. TABLES
3.1 GF(2)
3.2 GF(3)
3.3 GF(4)
3.4 GF(5)
3.5 GF(7)

### 1. INTRODUCTION

Let GF(q) be the Galois field with q elements and o be an automorphism of GF(q). A bilinear form on the vector space GF(q)n is a map f : GF(q)nxGF(q)n->GF(q) such that for all x,y,z in GF(q)n and all a in GF(q), we have:

f(x+y,z) = f(x,z)+f(y,z)        f(ax,z) = af(ax,z)
f(x,y+z) = f(x,y)+f(x,z)        f(x,az) = af(x,z).

The form f is symmetric if f is bilinear and f(x,y)=f(y,x) for all x,y in GF(q)n and hermitian symmetric if o² = id and f(x,y)=f(y,x)o. If f(x,y)=0 we say that x and y are orthogonal. For a subset S of GF(q)n we define the orthogonal of S as the subspace of GF(q)n formed with the elements x such that f(x,y)=0 for all y in S. We say that f is nondegenerate if the orthogonal of GF(q)n is the null space.

Example The usual inner product of vectors x=(x1,...,xn) and y=(y1,...,yn) in GF(q)n defined by

x.y= x1.y1+...+xn.yn
is a nondegenerate symmetric form called the euclidian scalar product.

A linear code of length n and dimension k is a k-dimensional subspace of GF(q)n. The elements of a code are its codewords. The Hamming weight of x=(x1,...,xn) in GF(q)n is the number of its non-zero coordinates and the minimum distance d(C) of a code C is the minimum of the non-zero weights of its codewords. A self-dual code is a code that is equal to its orthogonal.

For those who desire more informations should refer to the the chapter "Self-dual codes" written by E.M. Rains and N.J.A. Sloane in Handbook of Coding Theory edited by V.S. Pless and W.C. Huffman, (Amsterdam: Elsevier, 1998, pp. 177-294).

### 3.1 GF(2)

GF(2)n is equipped with the euclidian scalar product. You can find an up to date table of self-dual binary codes with the highest known minimum distance here

### 3.2 GF(3)

GF(3)n is equipped with the euclidian scalar product. You can find an up to date table of self-dual codes with the highest known minimum distance here

### 3.4 GF(5)

GF(5)n is equipped with the euclidian scalar product. You can find an up to date table of self-dual codes with the highest known minimum distance here

### 3.5 GF(7)

GF(7)n is equipped with the euclidian scalar product. You can find an up to date table of self-dual codes with the highest known minimum distance here