Core Concepts
There are only three possible families of completely regular self-dual codes with covering radius ρ = 2: the binary extended Hamming [8, 4, 4] code, the direct product of two ternary Hamming [4, 2, 3] codes, and a family of [4, 2, 3]q codes for q = 2^r where r > 1.
Abstract
The paper provides a complete classification of completely regular self-dual codes with covering radius ρ = 2. The key insights are:
There are only two sporadic such codes, of length 8, and an infinite family, of length 4.
For the length 8 codes:
The binary extended Hamming [8, 4, 4] code is self-dual, completely regular, and antipodal.
The [8, 4, 3]3 code is the direct product of two ternary Hamming [4, 2, 3]3 codes.
For the length 4 family, the [4, 2, 3]q codes are self-dual, completely regular, and antipodal, where q = 2^r and r > 1.
The authors provide a detailed description of all these codes and give the intersection arrays for all of them.