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Divisible minimal codes

Title data

Chubenko, Vladimir ; Kurz, Sascha:
Divisible minimal codes.
Bayreuth , 2025 . - 21 p.
DOI: https://doi.org/10.15495/EPub_UBT_00007346

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Official URL: Volltext

Abstract in another language

Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a k-dimensional linear code over GF(q) is denoted by m(k,q). Here we determine m(7,2), m(8,2), and m(9,2), as well as full classifications of all codes attaining m(k,2) for k<=7 and those attaining m(9,2). For m(11,2) and m(12,2) we give improved upper bounds. It turns out that in many cases attaining extremal codes have the property that the weights of all codewords are divisible by some constant &\Delta;>1. So, here we study the minimum lengths of minimal codes where we additionally assume that the weights of the codewords are divisible by &\Delta;.

Abstract in another language

Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. %he minimum possible length of such a k-dimensional linear code over GF(q) is denoted by m(k,q). Here we determine m(7,2), m(8,2), and m(9,2), as well as full classifications of all codes attaining m(k,2) for k at most 7 and those attaining m(9,2). We give improved upper bounds for m(k,2) for all k between 10 and 17. It turns out that in many cases the attaining extremal codes have the property that the weights of all codewords are divisible by some constant. So, here we study the minimum lengths of minimal codes where we additionally assume that the weights of the codewords are divisible by some given constant. As a byproduct we also give a few binary linear codes improving the best known lower bound for the minimum distance.

Further data

Item Type: Preprint, postprint
Keywords: minimal codes; divisible codes; quasi-cyclic codes; optimal codes; acute sets
Subject classification: Mathematics Subject Classification Code: 94B05 (51E23)
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 Mathematical Economics
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: 12 May 2025 06:01
Last Modified: 06 Oct 2025 12:07
URI: https://eref.uni-bayreuth.de/id/eprint/93495

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