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New lower bounds on q-ary error-correcting codes

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Abstract

Let Aq (n, d) denote the maximum size of a q-ary code with length n and minimum distance d. For most values of n and d, only lower and upper bounds on Aq (n, d) are known. In this paper new lower bounds on and updated tables of Aq (n, d) for q ∈ {3, 4, 5} are presented. The new bounds are obtained through an extensive computer search for codes with prescribed groups of automorphisms. Groups that act transitively on the (coordinate,value) pairs as well as groups with certain other closely related actions are considered.

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Acknowledgments

Electronic tables of bounds maintained by Andries Brouwer have been of great help to the authors.

Author information

Authors and Affiliations

  1. Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 15400, 00076, Aalto, Finland

    Antti Laaksonen & Patric R. J. Östergård

Authors
  1. Antti Laaksonen
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  2. Patric R. J. Östergård
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Corresponding author

Correspondence to Antti Laaksonen.

Additional information

This article is part of the Topical Collection on Special Issue on Coding Theory and Applications

This work was supported in part by the Academy of Finland, Project #289002.

Appendix: Codes Attaining New Lower Bounds

Appendix: Codes Attaining New Lower Bounds

Bound: :

A5 (7, 5) ≥ 57

Generators of G : :

(2 9)(3 17)(4 11)(5 26)(6 27)(7 21)(10 24)(12 19)(13 20)(14 28)(16 23)(18 25), (1 3 5 14 23 25 27)(2 4 6 22 24 26 21)(7 16 18 20 8 10 12) (9 11 13 15 17 19 28)(29 31 33 35 30 32 34)

Orbit representatives: :

0304000, 4444444

Bound: :

A5 (8, 5) ≥ 257 (1 14 9 6)(2 21 26 13)(3 8 19 24)(4 15)(5 10 29 18)(7 12) >(11 16 27 32)(17 30 25 22)(20 31)(23 28)(33 38)(34 37)(35 40)(36 39), (1 28 17 4)(2 27)(3 26)(5 16 21 32)(6 7 14 15)(8 29 24 13) (9 20 25 12)(10 19)(11 18)(22 31 30 23)(33 36)(34 35)(37 40)(38 39), (1 32 25 16)(2 15 10 7)(3 14 27 30)(4 5)(6 19 22 11)(8 9 24 17) (12 13)(18 31 26 23)(20 21)(28 29)(33 40)(34 39)(35 38)(36 37)

Orbit representatives: :

21401000, 44444444

Bound: :

A5 (9, 5) ≥ 857

Generators of G : :

(1 30 24 14)(2 35 25 36)(3 33 23 19)(4 13)(5 10 12 6)(7 9 11 8) (15 32 28 21)(16 18 29 17)(20 26 34 27)(22 31)(37 39 42 41)(38 44 43 45), (1 11 34)(2 7 28)(3 22 17)(4 8 30)(5 9 33)(6 32 27)(10 29 16)(12 31 35) (13 26 21)(14 18 24)(15 23 36)(19 20 25)(37 38 43)(39 40 44)(41 45 42)

Orbit representatives: :

114040000, 312431000, 221012000, 443423000, 334241100, 000322200, 224143300, 401410010, 024433010, 023240110, 031121110, 112402410, 444444444

Bound: :

A5 (9, 6) ≥ 157

Generators of G : :

(1 3 2)(4 31 22)(5 32 23)(6 33 24)(7 35 18)(8 36 16)(9 34 17) (10 30 20)(11 28 21)(12 29 19)(25 26 27)(37 39 38)(43 44 45), (1 18 32 2 16 31 3 17 33)(4 21 26 6 19 27 5 20 25)(7 13 30 8 15 28 9 14 29) (10 36 23 11 34 22 12 35 24)(37 45 41 38 43 40 39 44 42)

Orbit representatives: :

111000000, 412423100, 144310210, 444444444

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Laaksonen, A., Östergård, P.R.J. New lower bounds on q-ary error-correcting codes. Cryptogr. Commun. 11, 881–889 (2019). https://doi.org/10.1007/s12095-018-0302-9

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