A survey of self-dual codes, written for the Handbook of Coding Theory. Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this chapter include codes over F2, F3, F4, Fq, Z4, Zm, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.

Self-Dual Codes

The alphabet F/sub 2/+uF/sub 2/ is viewed here as a quotient of the Gaussian integers by the ideal (2). Self-dual F/sub 2/+uF/sub 2/ codes with Lee weights a multiple of 4 are called Type II. They give even unimodular Gaussian lattices by Construction A, while Type I codes yield unimodular Gaussian lattices. Construction B makes it possible to realize the Leech lattice as a Gaussian lattice. There is a Gray map which maps Type II codes into Type II binary codes with a fixed point free involution in their automorphism group. Combinatorial constructions use weighing matrices and strongly regular graphs. Gleason-type theorems for the symmetrized weight enumerators of Type II codes are derived. All self-dual codes are classified for length up to 8. The shadow of the Type I codes yields bounds on the highest minimum Hamming and Lee weights.

Type II codes over F/sub 2/+uF/sub 2/

Necessary and sufficient conditions are presented for two doubly even codes with an automorphism of odd prime order to be equivalent under certain assumptions for the codes. As an application we obtain all extremal doubly even codes of length 56 with an automorphism of order 13 . Sixteen such codes are inequivalent.

A method for constructing inequivalent self-dual codes with applications to length 56

We prove that there does not exist a [72,36,16] doubly even code with an automorphism of order 11 . If such a code exists, it can be decomposed as a direct sum of two codes; one can be viewed as a self-dual [12,6,4] binary code and the other a self-dual [6,3,4] code over F_{1024} . This is shown to be impossible.

A [72,36,16] doubly even code does not have an automorphism of order 11 (Corresp.)

Extremal doubly-even [80,40,16] codes with an automorphism of order 19

An interesting open question is whether a (72, 36, 16) doubly even code C exists In [3] the odd prime numbers which can divide the order of the group of C were determined and 23 is the largest of these Twenty-three is eliminated by reducing the problem to the consideration of 348 codes, each of which is shown to have minimum weight 12 or less One of these codes, denoted by C' , arises from the (a + x, b + x, a + b + x) construction where a and b are in one quadratic residue code and x is in the other The weight distribution of C' is given

23 does not divide the order of the group of a (72,36,16) doubly even code (Corresp.)

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23 does not divide the order of the group of a (72,36,16) doubly even code (Corresp.)