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1.Table of general 5-ary codes

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

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