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On strongly walk regular graphs, triple sum sets and their codes

Title data

Kiermaier, Michael ; Kurz, Sascha ; Solé, Patrick ; Stoll, Michael ; Wassermann, Alfred:
On strongly walk regular graphs, triple sum sets and their codes.
In: Designs, Codes and Cryptography. Vol. 91 (2023) . - pp. 645-675.
ISSN 1573-7586
DOI: https://doi.org/10.1007/s10623-022-01118-z

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Abstract in another language

Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an s-SWRG has at most 4 distinct eigenvalues k>θ1>θ2>θ3, and that the triple (θ1,θ2,θ3) satisfies a certain homogeneous polynomial equation of degree s−2 (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for s=5 and s=7, there are only the obvious solutions, and we conjecture this to remain true for all (odd) s≥9.

Further data

Item Type: Article in a journal
Refereed: Yes
Keywords: strongly walk-regular graphs; triple sum sets; three-weight codes;
Subject classification: Mathematics Subject Classification Code: 05E30 (11D41 94B05)
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra)
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) > Chair Mathematics II (Computer Algebra) - Univ.-Prof. Dr. Michael Stoll
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematical Economics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics and Didactics
Faculties
Result of work at the UBT: Yes
DDC Subjects: 000 Computer Science, information, general works > 004 Computer science
500 Science > 510 Mathematics
Date Deposited: 17 Aug 2023 07:30
Last Modified: 01 Dec 2023 08:36
URI: https://eref.uni-bayreuth.de/id/eprint/86589

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