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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 179 64 9
6lb 164 = =
7ub 614 169 32 8
7lb 512 128 = =
8ub 2340 611 128 32 5
8lb 2048 320 70 = =
9ub 9344 2314 512 128 20 5
9lb 8192 1024 256 64 18 =
10ub 30427 8951 2045 496 80 16 5
10lb 17408 4096 1024 256 40 = =
11ub 109226 30427 6241 1780 320 64 12 4
11lb 65536 16384 4096 1024 128 34 = =
12ub 419430 109226 20852 5864 1167 242 48 9 4
12lb 262144 65536 8192 4096 256 128 26 = =

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 subsequent improvements:

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).

Improvements are welcome.

Andries Brouwer - aeb@cwi.nl


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