The representation of any affine plane of prime-power order within its own line-code (Q1304828)

The paper under review is based on the theory of cone-representations due to the author [see Bull. Inst. Math. Soc. 49, No. 2, 311-324 (1994; Zbl 0799.05008) and Boll. Unione Mat. Ital., VII. Ser. A 11, No. 3, 789-800 (1997; Zbl 0901.51006)]. Let \(\pi\) be an affine plane of order \(q\), where \(q\) is some prime power. Let \(V\) be the vector space of all maps from the points of \(\pi\) to GF(\(q)\). The line code \(L\) of \(\pi\) is the subspace of \(V\) which is generated by the characteristic maps of the lines of \(\pi\). The \(q+1\) parallel classes of lines of \(\pi\) correspond to subspaces \(S_j\), \(j=1,\dots,q+1\), of \(\text{PG}(L)\). A vector \(\alpha = (\alpha_1,\dots,\alpha_{q+1}) \in V\) with \(\alpha_j \in S_j\) is called a substitution. A substitution \(\alpha\) is said to be faithful, if \(\{\alpha_{jk}\mid k=1,\dots,q\} = \text{GF}(q)\) for any \(j\), where \(\alpha_j = \sum_{k=1}^{q+1}\alpha_{jk}\chi_{jk}\) and the \(\chi_{jk}\) are the characteristic maps of the lines of the parallel class \(S_j\). The author proves that for any faithful substitution \(\alpha\) there exists a natural cone-representation \(R(\alpha)\) of index one of the dual of \(\pi\) in the projective space \(\text{PG}(L)\). Moreover, there exist \((q-1)q^{q+1}\) faithful substitutions that give rise to the same representation as \(\alpha\).