Results on error-correcting codes

Let A_q(n,d) denote the maximum size of a q-ary code with length n and minimum distance d. We present several new lower bounds on A_q(n,d) where q ∈ {2,3,4,5}.

1. Results on binary codes

Laaksonen, A., Östergård, P. R. J.: Constructing error-correcting binary codes using transitive permutation groups. Discrete Applied Mathematics 233, 65–70 (2017)

• A₂(18,4) ≥ 5632
• A₂(21,4) ≥ 40960
• A₂(22,4) ≥ 81920
• A₂(23,4) ≥ 163840
• A₂(24,4) ≥ 327680
• A₂(24,10) ≥ 136
• A₂(25,6) ≥ 17920

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2. Results on ternary, quaternary and quinary codes

Laaksonen, A., Östergård, P. R. J.: New lower bounds on error-correcting ternary, quaternary and quinary codes. In: Proceedings of the 5th International Castle Meeting on Coding Theory and Applications, 228–237 (2017)

• A₃(13,4) ≥ 13122
• A₃(14,4) ≥ 27702
• A₃(15,4) ≥ 83106
• A₃(15,5) ≥ 7812
• A₃(15,6) ≥ 3321
• A₃(16,7) ≥ 1026
• A₃(16,8) ≥ 387
• A₄(8,4) ≥ 352
• A₄(8,5) ≥ 76
• A₄(9,4) ≥ 1152
• A₄(9,6) ≥ 76
• A₄(10,3) ≥ 24576
• A₄(10,4) ≥ 4192
• A₄(11,3) ≥ 77056
• A₅(8,4) ≥ 1225
• A₅(8,5) ≥ 165
• A₅(9,4) ≥ 4375
• A₅(9,5) ≥ 725
• A₅(10,4) ≥ 17500

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Additional (unpublished) results on quinary codes:

• A₅(7,4) ≥ 257
• A₅(7,5) ≥ 57
• A₅(8,5) ≥ 257
• A₅(9,5) ≥ 857
• A₅(9,6) ≥ 157

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