Self-Dual Code

A self-dual code is a linear code equal to its dual code under the standard dot product. Consequently . This is Euclidean self-duality (MacWilliams and Sloane 1977).

A binary code that is self-dual is doubly even when every codeword has Hamming weight divisible by four. Otherwise it is singly even. At length 60, an extremal binary self-dual code has minimum distance 12, attaining the upper bound for binary self-dual codes, and has parameters , where the three numbers give length, dimension, and minimum Hamming distance (Rains 1998).

Harada (2026) completed the determination of the possible weight enumerators for these codes. In the family beginning

a code exists precisely for . The new constructions supply the previously missing values 3, 4, and 8. This determines possible enumerators, rather than all equivalence classes of codes. Harada (2026) reports GPT-6 Astra assistance with the constructions, which he checked independently in Magma.