On the nonexistence of ternary linear codes attaining the Griesmer bound

An [ n , k , d ] q code is a linear code of length n , dimension k and minimum weight d over the field of order q . It is known that the Griesmer bound is attained for all sufficiently large d for fixed q and k . We deal with the problem to find D q , k , the largest value of d such that the Griesme...

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Bibliographic Details
Published inDesigns, codes, and cryptography Vol. 90; no. 4; pp. 947 - 956
Main Authors Kawabata, Daiki, Maruta, Tatsuya
Format Journal Article
LanguageEnglish
Published New York Springer US 2022
Springer Nature B.V
Subjects
Codes
Coding and Information Theory
Computer Science
Cryptology
Discrete Mathematics in Computer Science
Minimum weight
Geometric method
Griesmer bound
Optimal linear codes
51E20
94B05
94B27
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Summary:An [ n , k , d ] q code is a linear code of length n , dimension k and minimum weight d over the field of order q . It is known that the Griesmer bound is attained for all sufficiently large d for fixed q and k . We deal with the problem to find D q , k , the largest value of d such that the Griesmer bound is not attained for fixed q and k . D q , k is already known for the cases q ≥ k with k = 3 , 4 , 5 and q ≥ 2 k - 3 with k ≥ 6 , but not known for the case q < k except for some small q and k . We show that our conjecture on D 3 , k is valid for k ≤ 9 .
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-022-01021-7