Many More Names of (7, 3, 1)

This design, which we call the (7, 3, 1) design, makes appearances in many areas of mathematics. It seems to turn up again and again in unexpected places. An earlier paper in this MAGAZINE [4] described (7, 3, 1)’s appearance in a number of different areas, including finite projective planes, as the Fano plane (FIGURE 1); graph theory, as the Heawood graph and the doubly regular round-robin tournament of order 7; topology, as an arrangement of six mutually adjacent hexagons on the torus; (−1, 1) matrices, as a skew-Hadamard matrix of order 8; and algebraic number theory, as the splitting field of the polynomial (x2 − 2)(x2 − 3)(x2 − 5). In this paper, we show how (7, 3, 1) makes appearances in three areas, namely (1) Hamming’s error-correcting codes, (2) Singer designs and difference sets based on n-dimensional finite projective geometries, and (3) normed algebras. We begin with an overview of block designs, including two descriptions of (7, 3, 1). We then describe certain binary error-correcting codes called Hamming codes, in which (7, 3, 1) makes three different appearances. Next, we expand the treatment of Hamming codes from binary codes to codes over all finite fields Fq , where q is an odd prime. Then, we describe generalizations of the block design structure of (7, 3, 1) to