On metric regularity of Reed–Muller codes

In this work we study metric properties of the well-known family of binary Reed–Muller codes. Let A be an arbitrary subset of the Boolean cube, and \({\widehat{A}}\) be the metric complement of A—the set of all vectors of the Boolean cube at the maximal possible distance from A. If the metric complement of \({\widehat{A}}\) coincides with A, then the set A is called a metrically regular set. The problem of investigating metrically regular sets appeared when studying bent functions, which have important applications in cryptography and coding theory and are also one of the earliest examples of a metrically regular set. In this work we describe metric complements and establish the metric regularity of the codes \({{\mathcal {R}}}{{\mathcal {M}}}(0,m)\) and \({{\mathcal {R}}}{{\mathcal {M}}}(k,m)\) for \(k \geqslant m-3\). Additionally, the metric regularity of the codes \({{\mathcal {R}}}{{\mathcal {M}}}(1,5)\) and \({{\mathcal {R}}}{{\mathcal {M}}}(2,6)\) is proved. Combined with previous results by Tokareva (Discret Math 312(3):666–670, 2012) concerning duality of affine and bent functions, this establishes the metric regularity of most Reed–Muller codes with known covering radius. It is conjectured that all Reed–Muller codes are metrically regular.