The table below gives upper and lower bounds for A4(n,d), the maximum number of vectors in a quaternary 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 4.
If d = 1 then this maximum is 4^n.
If d = 2 then this maximum is 4^(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 | ||
| 4 | 16 | ||||||||||
| 5 | 64 | 16 | |||||||||
| 6ub | 176 | 64 | 9 | ||||||||
| 6lb | 164 | = | = | ||||||||
| 7ub | 596 | 155 | 32 | 8 | |||||||
| 7lb | 512 | 128 | = | = | |||||||
| 8ub | 2340 | 611 | 128 | 32 | 5 | ||||||
| 8lb | 2048 | 352 | 76 | = | = | ||||||
| 9ub | 9344 | 2314 | 512 | 120 | 20 | 5 | |||||
| 9lb | 8192 | 1152 | 256 | 76 | 18 | = | |||||
| 10ub | 30427 | 8951 | 2045 | 480 | 80 | 16 | 5 | ||||
| 10lb | 24576 | 4192 | 1024 | 256 | 48 | = | = | ||||
| 11ub | 109226 | 30427 | 6241 | 1780 | 320 | 60 | 12 | 4 | |||
| 11lb | 77056 | 16384 | 4096 | 1024 | 128 | 48 | = | = | |||
| 12ub | 419430 | 109226 | 20852 | 5864 | 1167 | 240 | 48 | 9 | 4 | ||
| 12lb | 262144 | 65536 | 8192 | 4096 | 256 | 128 | = | = | = | ||
The table above is taken from
Galina T. Bogdanova, Andries E. Brouwer, Stoian N. Kapralov & Patric R.J. Östergård, Error-Correcting Codes over an Alphabet of Four Elements, Designs, Codes and Cryptography 23 (2001) 333-342.
with the following improvements:
A4(12,9) = 48 (Conrad Mackenzie and Jennifer Seberry, Maximal q-ary codes and Plotkin's Bound, Ars Combinatoria 26B (1988) 37-50).
A4(12,8) ≥ 128, A4(11,7) ≥ 128. (A. E. Brouwer, Small additive quaternary codes, preprint 2002.)
A4(7,4) ≤ 169, A4(8,4) ≤ 611, A4(9,4) ≤ 2314, A4(10,4) ≤ 8951, A4(10,5) ≤ 2045, A4(10,6) ≤ 496, A4(11,6) ≤ 1780, A4(12,6) ≤ 5864, A4(12,7) ≤ 1167. (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.)
A4(9,3) ≤ 9344 (W. Lang, J. Quistorff, E. Schneider, New Results on Integer Programming for Codes, preprint, 2007).
A4(11,8) ≥ 34, A4(12,9) ≥ 26 (Peter Andrews, pers.comm., 2011).
A4(6,3) ≤ 176, A4(7,3) ≤ 596, A4(7,4) ≤ 155 (Bart Litjens, Sven Polak & Alexander Schrijver, Semidefinite bounds for nonbinary codes based on quadruples, Des. Codes Cryptogr. online May 2016).
A4(9,6) ≤ 120, A4(11,8) ≤ 60 (Sven Polak, New non-binary code bounds based on a parity argument, arXiv:1606.05144).
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 (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