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