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1.Table of general ternary codes

The table below gives upper and lower bounds for A3(n,d), the maximum number of vectors in a ternary code of word length n and with Hamming distance d.

If d > n then this maximum is 1.
If d = n then this maximum is 3.
If d = 1 then this maximum is 3^n.
If d = 2 then this maximum is 3^(n-1).

Thus, in the table below we may restrict ourselves to the cases 2 < d < n. Horizontally we give d, vertically n. The `ub' rows give upper bounds, the `lb' rows lower bounds, and an `=' entry means that upper bound equals lower bound so that the value is exact.

 3 4 5 6 7 8 9 10 11 12 13 14 15 4 9 5 18 6 6 38 18 4 7ub 111 33 10 3 7lb 99 = = = 8ub 333 99 27 9 3 8lb 252 = = = = 9ub 937 297 81 27 6 3 9lb 729 243 = = = = 10ub 2808 891 243 81 14 6 3 10lb 2187 729 = = = = = 11ub 7029 2561 729 243 36 12 4 3 11lb 6561 1458 = = = = = = 12ub 19683 6839 1557 729 108 36 9 4 3 12lb = 4374 729 = 60 = = = = 13ub 59049 19270 4078 1449 324 95 27 6 3 3 13lb = 13122 2187 729 162 54 = = = = 14ub 153527 54774 10624 3660 805 237 62 13 6 3 3 14lb 118098 27702 6561 2187 243 108 36 = = = = 15ub 434815 149585 29213 9904 2204 685 165 39 10 6 3 3 15lb 354294 83106 7812 3321 729 243 81 24 = = = = 16ub 1240029 424001 77217 27356 6235 1923 451 114 29 9 4 3 3 16lb 1062882 216513 19683 6561 1026 387 243 54 18 = = = =

The table above is the one from A.E. Brouwer, H.O. Hämäläinen, P.R.J. Östergård & N.J.A. Sloane, Bounds on Mixed Binary/Ternary Codes, IEEE Trans. Inf. Th. 44 (1998) 140-161.

with the following subsequent improvements:

A3(6,3) = 38, A3(7,3) ≤ 111 and hence A3(8,3) ≤ 333. (P.R.J. Östergård, Classification of binary/ternary one-error-correcting codes, Discrete Math. 223 (2000) 253-262.)

A3(7,4) = 33 and hence A3(8,4) = 99, A3(9,4) ≤ 297, A3(10,4) ≤ 891. (P.R.J. Östergård, On binary/ternary error-correcting codes with minimum distance 4, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (M. Fossorier, H. Imai, S. Lin, and A. Poli, Eds.), LNCS 1719, Springer, Berlin 1999, pp. 472-481.)

A3(8,3) ≥ 252 was found by `ehl555`.

A3(14,4) ≥ 24786 was found by `Código`. A3(13,4) ≥ 8559 was found by `Código`.

A3(10,7) = 14 and hence A3(11,7) ≤ 42, A3(12,7) ≤ 126. (K.S. Kapralov, The nonexistence of ternary (10,15,7) codes, Proc. seventh international workshop on algebraic and combinatorial coding theory (ACCT'2000), Bansko, Bulgaria, 18-24 June, 2000, pp. 189-192.)

A3(11,7) = 36 and hence A3(12,7) ≤ 108, A3(13,7) ≤ 324. Also, A3(14,10) = 13. (M.J. Letourneau & S.K. Houghten, Optimal Ternary (11,7) and (14,10) Codes, Journal of Combinatorial Mathematics and Combinatorial Computing 51 (2004) 159-164.)

A3(12,7) ≥ 54. (Kai Valinen, pers. comm., June 2002.)

A3(12,7) ≥ 60 was found by `spaik`.

A3(13,7) ≥ 162 was found by `spaik`.

A3(13,8) ≥ 54 was found by `PacoHH`.

A3(14,8) ≥ 108 was found by `Joan`.

A3(14,9) ≥ 36 was found by `spaik`.

A3(14,10) ≤ 13 and hence A3(15,10) ≤ 39, A3(16,10) ≤ 117. (P. Kaski & P.R.J. Östergård, There exists no (15,5,4) RBIBD, J. Combin. Des. 9 (2001) 227-232; reprinted in 9 (2001) 357-362.)

A3(12,4) ≤ 6839, A3(13,4) ≤ 19270, A3(14,4) ≤ 54774, A3(15,4) ≤ 149585, A3(16,4) ≤ 424001, A3(12,5) ≤ 1557, A3(13,5) ≤ 4078, A3(14,5) ≤ 10624, A3(15,5) ≤ 29213, A3(13,6) ≤ 1449, A3(14,6) ≤ 3660, A3(15,6) ≤ 9904, A3(16,6) ≤ 27356, A3(14,7) ≤ 805, A3(15,7) ≤ 2204, A3(16,7) ≤ 6235, A3(13,8) ≤ 95, A3(15,8) ≤ 685, A3(16,8) ≤ 1923, A3(14,9) ≤ 62, A3(15,9) ≤ 165, A3(16,10) ≤ 114. (Dion Gijswijt, Alexander Schrijver, Hajime Tanaka, New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming, preprint, 2004; JCT (A) 113 (2006) 1719-1731.)

A3(10,3) ≤ 2808, A3(16,3) ≤ 1304424 (W. Lang, J. Quistorff, E. Schneider, New Results on Integer Programming for Codes, preprint, 2007).

A3(16,3) ≤ 1240029 (E. Bellini, E. Guerrini, M. Sala, Some bounds on the size of codes, IEEE Trans. Inf. Th. 60 (2014) 1475-1480.)

A3(16,11) ≤ 29 (Sven Polak, New non-binary code bounds based on a parity argument, arXiv:1606.05144).

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 (Antti Laaksonen & Patric R. J. Östergård, New Lower Bounds on Error-Correcting Ternary, Quaternary and Quinary Codes, pp 228-237 in: International Castle Meeting on Coding Theory and Applications ICMCTA 2017, Lecture Notes in Computer Science 10495, Springer, 2017).

Improvements are welcome.

Andries Brouwer - aeb@cwi.nl

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