Codes of Steiner triple and quadruple systems (Q1802189)

Let \({\mathcal D} = ({\mathcal P},{\mathcal B})\) be an incidence structure with point set \(\mathcal P\) of order \(\nu\) and with block set \(\mathcal B\). The code \(C_ p({\mathcal D})\) (of length \(\nu\)) of \(\mathcal D\) over a prime field \(F_ p\) is the subspace of the space \(F^{\mathcal P}_ p\) of all functions from \(\mathcal P\) to \(F_ p\) that is spanned by incidence vectors of the blocks of \(\mathcal D\). If every block \(\mathcal D\) is incident with precisely \(k\) points and set of \(t\) distinct points are together incident with precisely \(\lambda\) blocks, then \(\mathcal D\) is said to be \(t-(\nu,k,\lambda)\) design. A Steiner triple (quadruple) system is any \(2(\nu,3,1)\) (resp. \(3- (\nu,4,1)\)) design. If \(C\) is a code of length \(\nu\) and every codeword of \(C\) has coordinate 0 at a particular coordinate position, then the code obtained from \(C\) by deleting this coordinate position is said to be a shortened code of \(C\). Let \(d\) to be an integer such that \(2^ d-1 \leq \nu < 2^{d+1} - 1\). The authors show that if \(\mathcal D\) is a Steiner triple system, then the binary code \(C_ 2({\mathcal D})\) of \(\mathcal D\) contains a subcode that can be shortened to the binary Hamming code \({\mathcal K}_ d\) of length \(2^ d- 1\). Similarly the binary code of any Steiner quadruple system on \(\nu + 1\) points contains a subcode that can be shortened to the Reed-Muller code \({\mathcal R}(d-2,d)\) of order \(d-2\) and length \(2^ d\), where \(d\) is as above. These results are analogous with a result obtained by \textit{J. Doyen, X. Hubaut} and \textit{M. Vandensavel} in Math. Z. 163, 251-259 (1978; Zbl 0373.05011).