Optimal binary constant weight codes and affine linear groups over finite fields

Abstract

The affine linear group of degree one, $\text {AGL}(1,\mathbb {F}_q)$, over the finite field $\mathbb {F}_q$, acts sharply two-transitively on $\mathbb {F}_q$. Given $S<\text {AGL}(1,\mathbb {F}_q)$ and an integer k, $1\le k\le q$, does there exist a k-element subset $B\subset \mathbb {F}_q$ whose set-wise stabilizer is S? Our main result is the derivation of two formulas which provide an answer to this question. This result allows us to determine all possible parameters of binary constant weight codes that are constructed from the action of $\text {AGL}(1,\mathbb {F}_q)$ on $\mathbb {F}_q$ to meet the Johnson bound. Consequently, for many parameters, we are able to determine the values of the function $A_2(n,d,w)$, which is the maximum number of codewords in a binary constant weight code of length n, weight w and minimum distance $\ge d$.

Appendix

1.1 A1. A Mathematica code for computing ${\mathcal {N}}(S(\gamma ^{(q-1)/d},0,H),k)$

In the following Mathematica program, the input is $q=p^\alpha$; the output is the array $(k,d,o_d(p),i,j,\beta ,{\mathcal {N}})$, where

$$\begin{aligned}&0\le k\le p^\alpha /2,\\&d\mid p^\alpha -1,\\&i\mid (\alpha /o_d(p)),\\&{\left\{ \begin{array}{ll} j=0,1&{}\text {if}\ i=\alpha /o_d(p),\\ 0<j<\alpha /o_d(p)i&{}\text {if}\ i<\alpha /o_d(p), \end{array}\right. }\\&\beta =o_d(\alpha )ij,\\&k\equiv 0\ \text {or}\ p^\beta \pmod {dp^\beta },\\&{\mathcal {N}}={\mathcal {N}}(S(\gamma ^{(q-1)/d},0,H),k), \text {where} |H|=p^\beta ,\; |H^\dagger |=p^{o_d(p)i}. \end{aligned}$$

(A1.1)

1.2 A2. Numerical results

Table 1 gives the values of ${\mathcal {N}}(S(\gamma ^{(q-1)/d},0,H),k)$ for $q\le 16$. To recall the meanings of and the conditions on the parameters, refer to (A1.1) in Appendix A1.

Table 1 Values of $\mathcal N=\mathcal N(S(\gamma ^{(q-1)/d},0,H),k)$, $q\le 16$