On linear ternary self-dual codes arising from strongly regular signed graphs

Document Type : Original paper

Author

Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11 000 Belgrade, Serbia

Abstract

Linear ternary self-dual codes generated by matrices of the form $[I \;\; A+xI]$ are considered, where $A$ is the adjacency matrix of a strongly regular signed graph and $x\in \mathbb{F}_3$. It is shown that, apart from a trivial exception in dimension four, every such code has distance at least six. Existence is established in the case where $A$ has exactly two distinct eigenvalues $\lambda$ and $\mu$ which must be irrational, and satisfy $\lambda=-\mu$ and $\lambda^2\equiv -1 \pmod 3$. Moreover, among complete and net-regular signed graphs, only the complete case may occur, and exclusively within the preceding two-eigenvalue setting, while the net-regular case is excluded entirely. In the end, the analysis reduces the problem to exactly two unresolved exceptional cases. Examples of the obtained codes are provided.

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References

[1] M. Andelić, T. Koledin, and Z. Stanić, Relationships between net regularity, strong regularity and walk regularity of signed graphs, Linear Multilinear Algebra 73 (2025), no. 6, 1062–1077. https://doi.org/10.1080/03081087.2024.2405044
[2] P.J. Cameron and J.H. van Lint, Designs, Graphs, Codes and Their Links, Cambridge University Press, Cambridge, 1991.
[3] M. Harada and A. Munemasa, A complete classification of ternary self-dual codes of length 24, J. Combin. Theory A 116 (2009), no. 5, 1063–1072. https://doi.org/10.1016/j.jcta.2008.11.011
[4] T. Koledin and Z. Stanić, On a class of strongly regular signed graphs, Publ. Math. Debrecen 97 (2020), 353–365. https://doi.org/10.5486/PMD.2020.8760
[5] J. Martínez-Bernal, M.A. Valencia-Bucio, and R.H. Villarreal, Linear codes over signed graphs, Des. Codes Cryptogr. 88 (2020), 273–296. https://doi.org/10.1007/s10623-019-00683-0
[6] V. Pless, Introduction to the Theory of Error-Correcting Codes, John Willey and Sons, New York, 1998.
[7] P. Solé and T. Zaslavsky, A coding approach to signed graphs, SIAM J. Discrete Math. 7 (1994), no. 4, 544–553. https://doi.org/10.1137/S0895480189174374
[8] Z. Stanić, On strongly regular signed graphs, Discrete Appl. Math. 271 (2019), 184–190. https://doi.org/10.1016/j.dam.2019.06.017
[9] Z. Stanić, Spectra of signed graphs with two eigenvalues, Appl. Math. Comput. 364 (2020), 124627. https://doi.org/10.1016/j.amc.2019.124627
[10] Z. Stanić, Signed graphs with two eigenvalues and vertex degree five, Ars Math. Contemp. 22 (2022), 1.10. https://doi.org/10.26493/1855-3974.2329.97a
[11] Z. Stanić, Linear ternary codes of strongly regular signed graphs, Discrete Math. 347 (2024), no. 1, 113714. https://doi.org/10.1016/j.disc.2023.113714
[12] Z. Stanić, Spectra of Signed Graphs, Cambridge University Press, Cambridge, 2026.
[13] J.H. van Lint, Introduction to Coding Theory, Springer, Berlin, 1999.
Communications in Combinatorics and Optimization
Volume 12, Issue 1
March 2027
Pages 327-338

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