The table below gives upper and lower bounds for
A_{3}(*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:

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

A_{3}(7,4) = 33 and hence
A_{3}(8,4) = 99, A_{3}(9,4) ≤ 297, A_{3}(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.)

A_{3}(8,3) ≥ 252 was
found
by `ehl555`

.

A_{3}(14,4) ≥ 24786 was
found
by `Código`

.
A_{3}(13,4) ≥ 8559 was
found
by `Código`

.

A_{3}(10,7) = 14 and hence
A_{3}(11,7) ≤ 42, A_{3}(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.)

A_{3}(11,7) = 36 and hence A_{3}(12,7) ≤ 108, A_{3}(13,7) ≤ 324.
Also, A_{3}(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.)

A_{3}(12,7) ≥ 54.
(Kai Valinen, pers. comm., June 2002.)

A_{3}(12,7) ≥ 60 was
found
by `spaik`

.

A_{3}(13,7) ≥ 162 was
found
by `spaik`

.

A_{3}(13,8) ≥ 54 was
found
by `PacoHH`

.

A_{3}(14,8) ≥ 108 was
found
by `Joan`

.

A_{3}(14,9) ≥ 36 was
found
by `spaik`

.

A_{3}(14,10) ≤ 13 and hence
A_{3}(15,10) ≤ 39, A_{3}(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.)

A_{3}(12,4) ≤ 6839, A_{3}(13,4) ≤ 19270, A_{3}(14,4) ≤ 54774,
A_{3}(15,4) ≤ 149585, A_{3}(16,4) ≤ 424001,
A_{3}(12,5) ≤ 1557, A_{3}(13,5) ≤ 4078, A_{3}(14,5) ≤ 10624, A_{3}(15,5) ≤ 29213,
A_{3}(13,6) ≤ 1449, A_{3}(14,6) ≤ 3660, A_{3}(15,6) ≤ 9904, A_{3}(16,6) ≤ 27356,
A_{3}(14,7) ≤ 805, A_{3}(15,7) ≤ 2204, A_{3}(16,7) ≤ 6235,
A_{3}(13,8) ≤ 95, A_{3}(15,8) ≤ 685, A_{3}(16,8) ≤ 1923,
A_{3}(14,9) ≤ 62, A_{3}(15,9) ≤ 165, A_{3}(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.)

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

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

A_{3}(16,11) ≤ 29
(Sven Polak, *New non-binary code bounds based on a parity argument*,
arXiv:1606.05144).

A_{3}(13,4) ≥ 13122, A_{3}(14,4) ≥ 27702, A_{3}(15,4) ≥ 83106,
A_{3}(15,5) ≥ 7812, A_{3}(15,6) ≥ 3321, A_{3}(16,7) ≥ 1026, A_{3}(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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