8th Croatian Mathematical Congress

Doubly even self-dual codes and even unimodular lattices constructed from Hadamard matrices

3 Jul 2024, 15:00

35m

D3 (School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek)

Invited lecture CDM: Combinatorics and Discrete Mathematics Invited lectures

Speaker

Andrea Švob (Faculty of Mathematics, University of Rijeka)

Description

A Hadamard matrix of order $n$ is a $n\times n$ $(-1,1)$-matrix $H$ such that $H{H}^{\mathrm{\top }}=n{I}_n$. In this talk, we are concerned with constructing doubly even self-dual binary codes from Hadamard matrices. More precisely, to a Hadamard matrix of order $8t$ we relate a doubly even self-dual binary code of length $8t$, and give explicit constructions of doubly even self-dual binary codes from skew-type Hadamard matrices and conference graphs. It is known that a doubly even self-dual binary code yields an even unimodular lattice. Consequently, this construction of skew-type Hadamard matrices gives us a series of even unimodular lattices of rank $2^{i+2}t$, $i$ a positive integer.

References

1. E. Bannai, S. T. Dougherty, M. Harada, M. Oura, Type II codes, even unimodular lattices, and invariant rings, IEEE Trans. Inform. Theory 45 (1999), 1194-1205.
2. D. Crnković, A. Švob, Constructing doubly even self-dual codes and even unimodular lattices from Hadamard matrices, Appl. Algebra Engrg. Comm. Comput., (2023). https://doi.org/10.1007/s00200-023-00615-5
3. T. Miezaki, Design-theoretic analogies between codes, lattices, and vertex operator algebras, Des. Codes Cryptogr. 89 (2021), 763-780.
4. G. Nebe, E. M. Rains, N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics, Vol. 17, Springer-Verlag, 2006.