3-(2 m , 2m–1, 2m–2–1) design for first order Reed-Muller code R(1, m)

Abstract

A binary linear code in with dimension k and minimum distance d is called an [n, k, d] code. For each positive integer m, the first order Reed-Muller code R(1, m) is a linear code [2 m , m+1, 2 m–1]. When m=3, an extremal doubly-even self-dual code of length 23=8 exists. In this paper, it is shown that the first order Reed-Muller code R(1, m) holds a 3-(2 m , 2 m–1, 2 m–2–1) design for m≥3. It is also shown that Steiner system holds for m=3.