An integer determinant approach to the length of Type II codes and the weights of shadows
Abstract
It is well known that the length n of a binary Type II code is divisible by $8$. The classical proof relies on Gleason's theorem on weight enumerators. Motivated by a question of Vera Pless, we give elementary proofs of this fact that avoid invariant theory. Our main tool is a congruence modulo $8$ for the determinant of the integer Gram matrix AAT of a binary orthogonal matrix A, obtained directly from the Leibniz expansion. It gives a short proof of a theorem of Janusz: the number of rows of weight congruent to 3 pmod 4 in a binary orthogonal matrix is divisible by $4$. From this single statement we derive both the length restriction for Type II codes and the theorem of Conway and Sloane that every vector in the shadow of a binary self-dual code of length n has weight congruent to n/2 pmod 4. We also give two short character-theoretic proofs of the length restriction and explain how all of these arguments detect the Arf invariant of a quadratic form.
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BibTeX
@misc{baek2026integer,
title = {{An integer determinant approach to the length of Type II codes and the weights of shadows}},
author = {Jae-Hyun Baek and Jon-Lark Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30809},
eprint = {2610.30809},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30809},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An integer determinant approach to the length of Type II codes and the weights of shadows
AU - Jae-Hyun Baek
AU - Jon-Lark Kim
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30809
UR - https://arxiv.org/abs/2610.30809
AB - It is well known that the length n of a binary Type II code is divisible by $8$. The classical proof relies on Gleason's theorem on weight enumerators. Motivated by a question of Vera Pless, we give elementary proofs of this fact that avoid invariant theory. Our main tool is a congruence modulo $8$ for the determinant of the integer Gram matrix AA^T of a binary orthogonal matrix A, obtained directly from the Leibniz expansion. It gives a short proof of a theorem of Janusz: the number of rows of weight congruent to 3 pmod 4 in a binary orthogonal matrix is divisible by $4$. From this single statement we derive both the length restriction for Type II codes and the theorem of Conway and Sloane that every vector in the shadow of a binary self-dual code of length n has weight congruent to n/2 pmod 4. We also give two short character-theoretic proofs of the length restriction and explain how all of these arguments detect the Arf invariant of a quadratic form.
ER -