On triply even binary codes
Koichi Betsumiya, Akihiro Munemasa
TL;DR
This paper explores the structure of triply even binary codes, demonstrating how they can be constructed from doubly even codes and classifying all maximal codes of length 48.
Contribution
It introduces a method to build triply even codes from doubly even codes and classifies all maximal triply even codes of length 48.
Findings
Maximal triply even codes of length 48 are exactly 10 up to equivalence.
Every such code can be constructed from two doubly even codes of length 24.
A new construction method for triply even codes from doubly even codes is established.
Abstract
A triply even code is a binary linear code in which the weight of every codeword is divisible by 8. We show how two doubly even codes of lengths m_1 and m_2 can be combined to make a triply even code of length m_1+m_2, and then prove that every maximal triply even code of length 48 can be obtained by combining two doubly even codes of length 24 in a certain way. Using this result, we show that there are exactly 10 maximal triply even codes of length 48 up to equivalence.
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