On linear codes whose weights and length have a common divisor (Q876321)

The main result of this paper is the following. Let \(C\) be a \([n,k,d]\)-linear code over the finite field \(GF(q)\), \(q\) a power of a prime \(p\), and \(1<r<n\) an integer which divides \(n\) and all the weights of codes words. If the dual minimum distance is at least 3, then \(n\geq (r-1)q+(p-1)r\). This bound is sharp in some cases and a strong generalization of results in [\textit{S. Ball} and \textit{A. Blokhuis}, Proc. Am. Math. Soc. 126, No. 11, 3377--3380 (1998; Zbl 0914.51008), \textit{S. Ball, A. Blokhuis} and \textit{F. Mazzocca}, Combinatorics 17, 31--41 (1997; Zbl 0880.51003)]; a motivation comes from the so-called ``strong cylinder conjecture'' [\textit{T. Szönyi} and \textit{Z. Weiner}, J. Comb. Theory, Ser. A 95, No. 1, 88--101 (2001; Zbl 0983.51006)]. Now from the generator matrix of \(C\) we obtain a set \(S\) in \(PG(k-1,q)\) with the property that each hyperplane is incident with a multiple of \(r\) points of \(S\). Then they prove the aforementioned bound by using properties of certain polynomials over finite fields.