On the extended binary Hamming code as a minor of optimal self-dual codes

In this work we focus on minors of binary codes, especially in relation to self-dual binary codes with good error-correcting capabilities as measured by their distance. We show that every optimal binary code with distance at least 4, up to length 32, has an extended binary Hamming code minor. Furthermore, when restricted to the case of Type II codes, we show that all optimal codes beyond a certain length have the binary Hamming code and its dual as a minor. Finally, we cast the code minor problem in the setting of double-circulant codes as a substring problem, demonstrate how it can be used, and discuss the computational challenges involved.