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- W2995681693 abstract "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 {it metrically regular set}. The problem of investigating metrically regular sets appeared when studying {it 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 $mathcal{RM}(0,m)$ and $mathcal{RM}(k,m)$ for $k geqslant m-3$. Additionally, the metric regularity of the codes $mathcal{RM}(1,5)$ and $mathcal{RM}(2,6)$ is proved. Combined with previous results by Tokareva N. (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." @default.
- W2995681693 created "2019-12-26" @default.
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- W2995681693 date "2019-12-23" @default.
- W2995681693 modified "2023-09-27" @default.
- W2995681693 title "On metric regularity of Reed-Muller codes" @default.
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