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## 1.Table of general quaternary codes

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

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