On the nonexistence of certain M.D.S. codes and projective planes
The authors define an L(n,k,r) space as a set L of \(n^ r\) k-tuples over the alphabet \(\{\) 1,...,n\(\}\) such that any two distinct elements of L are at Hamming distance at least \(k-r+1\) from each other. They take \(2\leq r<k\). These spaces are also called MDS codes. Their main results concern the case \(k=n+r-1\), when L is a code of length \(n+r-1\), cardinality \(n^ r\), and distance at least n over an alphabet of n letters. In Theorem 4 they prove that if such a space \(L(n,n+r-1,r)\) exists with \(r\geq 3\) and \(n>2\), then \(n\equiv 0 (mod 4)\). For \(r=3\) they say ''(such spaces) may be regarded as a kind of coding extension of a finite affine or projective plane of order n'' and that there is therefore no such extension for a plane of order 10. In Theorem 5 they prove that if there is an \(L(n,n+r- 1,r)\) with \(r\geq 4\) and \(n\geq 6\), then \(n\equiv 0\) or 2 (mod 9). In combining these results with earlier results (collected in Theorem 1 from \textit{C. Maneri} and \textit{R. Silverman} [J. Algebra 4, 321-330 (1966; Zbl 0151.016); J. Comb. Theory, Ser. A 11, 118-121 (1971; Zbl 0291.05013); \textit{R. Silverman}, Can. J. Math. 12, 158-176 (1960; Zbl 0092.012)], they conclude that \(r\geq 4\) and \(n\geq 2\) imply that \(L(n,n+r-1,r)\) exists only if 36 divides n.
scientific article; zbMATH DE number 3904493