On equidistant binary codes of length \(n=4k+1\) with distance \(d=2k\) (Q753818)

The author gives a short proof of the following result: An equidistant binary code of blocklength \(4\ell +1\) and distance \(2\ell\) exists if and only if \(\ell\) is of the form \((u^ 2+u)\) and there exists a (v,k,\(\lambda\)) design (i.e., a projective design) with \(v=2u^ 2+2u+1\) and \(k=u^ 2\). Apparently the result was first proved by \textit{D. R. Stinson} and \textit{G. H. J. van Rees} [Combinatorica 4, 357-362 (1984; Zbl 0556.05010)]. The existence of such designs has been established for u a power of a prime.