Related papers: The Covering Radius of the Reed-Muller Code $RM(m-…

It was proved by J. Schatz that the covering radius of the second order Reed--Muller code $RM(2, 6)$ is 18 (IEEE Trans Inf Theory 27: 529--530, 1985). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In…

In 1981, Schatz proved that the covering radius of the binary Reed-Muller code $RM(2,6)$ is 18. For $RM(2,7)$, we only know that its covering radius is between 40 and 44. In this paper, we prove that the covering radius of the binary…

Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. We denote this code by $RM(m,r)$. The minimal distance of $RM(m,r)$ is $2^{m-r}$ and so it cannot correct more than half that…

We propose an effective version of the lift by derivation, an invariant that allows us to provide the classification of B(5,6,8)=RM(6, 8)/RM(4,8). The main consequence is to establish that the covering radius of the Reed-Muller RM(4,8) is…

We prove the covering radius of the third-order Reed-Muller code RM(3,7) is 20, which was previously known to be between 20 and 23 (inclusive). The covering radius of RM(3, 7) is the maximum third-order nonlinearity among all 7-variable…

The weight spectra of the Reed-Muller codes $RM(r,m)$ were unknown for $r=3,...,m-5$. In IEEE Trans. Inform. Theory 2024, Carlet determined the weight spectrum of $RM(m-5,m)$ for $m\ge10$ using the Maiorana-McFarland construction, where the…

We determine the weight spectra of the Reed-Muller codes $RM(m-3,m)$ for $m\ge 6$ and $RM(m-4,m)$ for $m\ge 8$. The technique used is induction on $m$, using that the sum of two weights in $RM(r-1,m-1)$ is a weight in $RM(r,m)$, and using…

The classical Reed-Muller codes over a finite field $\mathbb{F}_q$ are based on evaluations of $m$-variate polynomials of degree at most $d$ over a product set $U^m$, for some $d$ less than $|U|$. Because of their good distance properties,…

We give a polynomial time algorithm to decode multivariate polynomial codes of degree $d$ up to half their minimum distance, when the evaluation points are an arbitrary product set $S^m$, for every $d < |S|$. Previously known algorithms can…

We study generalized covering radii, a fundamental property of linear codes that characterizes the trade-off between storage, latency, and access in linear data-query protocols such as PIR. We prove lower and upper bounds on the generalized…

The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like…

This paper gives lower and upper bounds on the covering radius of codes over $\Z_{2^s}$ with respect to homogenous distance. We also determine the covering radius of various Repetition codes, Simplex codes (Type $\alpha$ and Type $\beta$)…

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives…

Polynomial representations of Boolean functions over various rings such as $\mathbb{Z}$ and $\mathbb{Z}_m$ have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including…

Reed-Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an…

We generalize to any q a theorem about covering radius of linear codes proved by Helleseth, Klove and Mykkelvit. Then we determine the covering radius of first order generalized Reed-Muller codes in second order generalized Reed-Muller…

This paper presents a method to determine a set of basis polynomials from the extended Euclidean algorithm that allows Generalized Minimum Distance decoding of Reed-Solomon codes with a complexity of O(nd).

This paper studies the parameters for which Reed-Muller (RM) codes over $GF(2)$ can correct random erasures and random errors with high probability, and in particular when can they achieve capacity for these two classical channels.…

A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The…

The largest Hamming distance between a Boolean function in $n$ variables and the set of all affine Boolean functions in $n$ variables is known as the covering radius $\rho_n$ of the $[2^n,n+1]$ Reed-Muller code. This number determines how…