The table below gives upper and lower bounds for A5(n,d), the maximum number of vectors in a 5-ary 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 5.
If d = 1 then this maximum is 5^n.
If d = 2 then this maximum is 5^(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 | ||
| 4 | 25 | ||||||||
| 5 | 125 | 25 | |||||||
| 6 | 625 | 125 | 25 | ||||||
| 7ub | 2291 | 489 | 87 | 15 | |||||
| 7lb | 1597 | 250 | 53 | = | |||||
| 8ub | 9672 | 2291 | 435 | 65 | 10 | ||||
| 8lb | 7985 | 1225 | 165 | 50 | = | ||||
| 9ub | 44642 | 9672 | 2152 | 325 | 50 | 10 | |||
| 9lb | 31040 | 4375 | 725 | 135 | = | = | |||
| 10ub | 217013 | 44642 | 9559 | 1625 | 250 | 50 | 7 | ||
| 10lb | 125000 | 17500 | 3125 | 625 | 125 | = | = | ||
| 11ub | 1085053 | 217013 | 44379 | 8125 | 1250 | 250 | 35 | 6 | |
| 11lb | 468750 | 78125 | 15625 | 3125 | 625 | 125 | 25 | = | |
The table above is taken from Galina T. Bogdanova & Patric R.J. Östergård, Bounds on Codes over an Alphabet of Five Elements, Discrete Mathematics 240 (2001) 13-19
with the following improvements:
A5(10,8) = 50 (Conrad Mackenzie and Jennifer Seberry, Maximal q-ary codes and Plotkin's Bound, Ars Combinatoria 26B (1988) 37-50).
A5(7,4) ≤ 545, A5(7,5) ≤ 108, A5(8,5) ≤ 485, A5(9,5) ≤ 2152, A5(10,5) ≤ 9559, A5(11,5) ≤ 44379, A5(10,6) ≤ 1855, A5(11,6) ≤ 8840 (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.)
A5(11,3) ≤ 1085065 (Jörn Quistorff, Improved Sphere Bounds in Finite Metric Spaces, Bull. Inst. Combin. Appl. 46 (2006) 69-80).
A5(11,3) ≤ 1085053 (W. Lang, J. Quistorff, E. Schneider, New Results on Integer Programming for Codes, preprint, 2007).
A5(10,8) ≥ 28 (Allen Poapst, email 2010-03-09).
A5(7,4) ≤ 489, A5(7,5) ≤ 87 (Bart Litjens, Sven Polak & Alexander Schrijver, Semidefinite bounds for nonbinary codes based on quadruples, Des. Codes Cryptogr. online May 2016).
A5(8,6) ≤ 65 (Sven Polak, New non-binary code bounds based on a parity argument, June 2016, arXiv:1606.05144).
A5(8,4) ≥ 1225, A5(8,5) ≥ 165, A5(9,4) ≥ 4375, A5(9,5) ≥ 725, A5(10,4) ≥ 17500 (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