Classification of Binary Constant Weight Codes

A binary code
C
⊆ F
₂
n
with minimum distance at least
d
and codewords of Hamming weight
w
is called an
(n
,
d
,
w
) constant weight code. The maximum size of an
(n
,
d
,
w
) constant weight code is denoted by
A
(
n
,
d
,
w
), and codes of this size are said to be optimal. In a computer-aided approach, optimal
(n
,
d
,
w
) constant weight codes are here classified up to equivalence for
d
=4,
n
≤ 12;
d
=6,
n
≤ 14;
d
=8,
n
≤ 17;
d
=10,
n
≤ 20 (with one exception);
d
=12,
n
≤ 23;
d
=14,
n
≤ 26;
d
=16,
n
≤ 28; and
d
=18,
n
≤ 28. Moreover, several new upper bounds on
A
(
n
,
d
,
w
) are obtained, leading among other things to the exact values
A
(12,4,5)=80,
A
(15,6,7)=69,
A
(18,8,7)=33,
A
(19,8,7)=52,
A
(19,8,8)=78, and
A
(20,8,8)=130 . Since
A
(15,6,6)=70, this gives the first known example of parameters for which
A
(
n
,
d
,
w
-1) >
A
(
n
,
d
,
w
) with
w
≤
n
/2. A scheme based on double counting is developed for validating the classification results.