Steiner systems $S(2,4,2^m)$ supported by a family of extended cyclic codes
Qi Wang
TL;DR
The paper addresses whether binary extended cyclic codes can support Steiner systems $S(2,4,2^m)$ for all even $m\ge4$. It adopts affine-invariant extended cyclic codes with generator $g_E$ to obtain $2$-designs via automorphism-group arguments and weight-distribution analysis, proving that for $m\equiv0\pmod{4}$ and $\gcd(m,e)=2$ the supports of weight-4 codewords yield a Steiner system $S(2,4,2^m)$. It further computes parameters of additional $2$-designs arising from higher weights and shows an isomorphism between these coding-theoretic designs and the classical affine-geometry Steiner systems $S(2,4,4^{m/2})$. Overall, the work extends Ding's results to all even $m$, providing a coding-theoretic realization of affine-geometry designs and enriching the link between extended cyclic codes and finite geometry.
Abstract
In [C. Ding, An infinite family of Steiner systems $S(2,4,2^m)$ from cyclic codes, {\em J. Combin. Des.} 26 (2018), no.3, 126--144], Ding constructed a family of Steiner systems $S(2,4,2^m)$ for all $m \equiv 2 \pmod{4}$ from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems $S(2,4,2^m)$ for all $m \equiv 0 \pmod{4}$ supported by a family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system $S(2,4,2^m)$ for all even $m \geq 4$. This paper also determines the parameters of other $2$-designs supported by this family of extended cyclic codes.