Some new distance-4 constant weight codes

Improved binary constant weight codes with minimum distance 4 
are constructed. A table with bounds on the chromatic number 
of small Johnson graphs is given.

Next we give a table with lower bounds for A(n, 4, 5) for 29 ≤ n ≤ 64, to be compared with the table in [18]. It improves all bounds from that paper except for the three values for n = 45, 46, 47. The values marked s are derived from Steiner systems S(5, 6, 36) and S(5, 6, 48) ( [1,5]). The values marked with a dot are exact.

The Partitioning Construction
A partition Π(n, w) = (C 1 , ..., C m ) is a partition of the set of all n w binary vectors of length n and weight w into codes C i that all have minimum distance at least 4. By definition, C j = ∅ for j > m.
The direct product Π(n 1 , w 1 ) × Π(n 2 , w 2 ) of two partitions (C 1 , ..., C m1 ) and (D 1 , ..., D m2 ) is the code i C i * D i (of word length n 1 + n 2 and weight w 1 + w 2 and size |C i |.|D i |), where for two codes C and D the code C * D is the code consisting of all possible concatenations c * d with c ∈ C and d ∈ D.
The partitioning construction for codes of length n, weight w and minimum distance 4 constructs the code C = i Π(n 1 , 2i + ) × Π(n 2 , w − 2i − ) where n = n 1 + n 2 and ∈ {0, 1} and the union is over all i with i ≥ 0 and 2i + ≤ w.
It is usually nontrivial to construct the required ingredients Π(n, w). However, for w ≤ 1 the partition is trivial, namely the partition into singletons, and for w = 2 the optimal partition is that of the n(n − 1)/2 pairs into n − 1 parts of size n/2 if n is even, and into n parts of size (n − 1)/2 if n is odd. Partitions Π(n, w) and Π(n, n − w) are related by complementation. It is always possible to find a Π(n, w) with at most n parts, cf. [12].

Improvements by Etzion & Bitan
The code C that results from the partitioning construction is not always maximal. Etzion & Bitan [9] gave a handful of examples of improvements. Let us redo two of their examples here (using improved ingredients).
For Table 2 we used only the obvious partitions: for w ≤ 3 the above ones, for w = 4 the Graham-Sloane partitions ( [3], Theorem 14), and finally for w = 5 the partition with one part as large as possible (the best lower bound known for A(n, 4, w)) and all other parts arbitrary, for example of size 1. It will be easy to improve these bounds a little.
For Table 1 we spent some effort to find good partitions. In Table 3 below we give the vector of part sizes for the partitions used. The actual partitions can be found near [2].  We give a partition Π(11, 4) with 10 parts 35 35 35 35 33 32 32 32 31 30 explicitly (in the notation of [3]).
In the case of J(10,4) there exists a coloring with 9 colors where the last color is used only once. So J(10,4) minus a vertex has chromatic number 8.
It would be interesting to give more general constructions for colorings of J(n, w) with fewer than n colors.