On those Boolean functions that are coset leaders of first order Reed-Muller codes

Abstract

In this paper, we study the class of those Boolean functions that are coset leaders of first order Reed-Muller codes. We study their properties and try to better understand their structure (which seems complex), by studying operations on Boolean functions that can provide coset leaders (we show that these operations all provide coset leaders when the operands are coset leaders, and that some can even produce coset leaders without the operands being coset leaders). We characterize those coset leaders that belong to the well known classes of direct sums of monomial Boolean functions and Maiorana-McFarland functions. Since all the functions of Hamming weight at most \(2^{n-2}\) are automatically coset leaders, we are interested in constructing infinite classes of coset leaders having possibly Hamming weight larger than \(2^{n-2}\).