相关论文: List-decoding homomorphism codes with arbitrary co…
In this paper, we establish the list-decoding capacity theorem for sum-rank metric codes. This theorem implies the list-decodability theorem for random general sum-rank metric codes: Any random general sum-rank metric code with a rate not…
We establish an equivalence between two important random ensembles of linear codes: random linear codes (RLCs) and random Reed-Solomon (RS) codes. Specifically, we show that these models exhibit identical behavior with respect to key…
Let $A(n, d)$ denote the maximum size of a binary code of length $n$ and minimum Hamming distance $d$. Studying $A(n, d)$, including efforts to determine it as well to derive bounds on $A(n, d)$ for large $n$'s, is one of the most…
In this paper show that the list and bounded-distance decoding problems of certain bounds for the Reed-Solomon code are at least as hard as the discrete logarithm problem over finite fields.
We give a complete classification of homomorphisms from the braid group on $n$ strands to the braid group on $2n$ strands when $n$ is at least 5. We also classify endomorphisms of the braid group on 4 strands, as well as homomorphisms from…
The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect…
We show that any q-ary code with sufficiently good distance can be randomly punctured to obtain, with high probability, a code that is list decodable up to radius $1 - 1/q - \epsilon$ with near-optimal rate and list sizes. Our results imply…
Codes on hypergraphs are an extension of the well-studied family of codes on bipartite graphs. Bilu and Hoory (2004) constructed an explicit family of codes on regular t-partite hypergraphs whose minimum distance improves earlier estimates…
We introduce the concepts of complex Grassmannian codes and designs. Let G(m,n) denote the set of m-dimensional subspaces of C^n: then a code is a finite subset of G(m,n) in which few distances occur, while a design is a finite subset of…
Two graphs $G$ and $H$ are homomorphism indistinguishable over a family of graphs $\mathcal{F}$ if for all graphs $F \in \mathcal{F}$ the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphism from $F$ to $H$. Many…
Locally Decodable Codes (LDCs) are error-correcting codes $C\colon\Sigma^n\rightarrow \Sigma^m,$ encoding \emph{messages} in $\Sigma^n$ to \emph{codewords} in $\Sigma^m$, with super-fast decoding algorithms. They are important mathematical…
A group homomorphism eta:A-> H is called a localization of A if every homomorphism phi:A-> H can be `extended uniquely' to a homomorphism Phi:H-> H in the sense that Phi eta = phi. This categorical concepts, obviously not depending on the…
We study the maximum Hamming distance (or rather, the complementary notion of "minimum approximability") of a general function on a finite group $G$ to either of the sets $\operatorname{End}(G)$ and $\operatorname{Aff}(G)$, of group…
In the present paper, we consider list decoding for both random rank metric codes and random linear rank metric codes. Firstly, we show that, for arbitrary $0<R<1$ and $\epsilon>0$ ($\epsilon$ and $R$ are independent), if $0<\frac{n}{m}\leq…
Neural codes are binary codes that are used for information processing and representation in the brain. In previous work, we have shown how an algebraic structure, called the {\it neural ring}, can be used to efficiently encode geometric…
In this paper we consider algebraic geometry (AG) codes: a class of codes constructed from algebraic codes (equivalently, using function fields) by Goppa. These codes can be list-decoded using the famous Guruswami-Sudan (GS) list-decoder,…
In this paper we present a minimal list decoding algorithm for Reed-Solomon (RS) codes. Minimal list decoding for a code $C$ refers to list decoding with radius $L$, where $L$ is the minimum of the distances between the received word…
We consider codes of length $m$ over an alphabet of size $q$ as subsets of the vertex set of the Hamming graph $\Gamma=H(m,q)$. A code for which there exists an automorphism group $X\leq Aut(\Gamma)$ that acts transitively on the code and…
Multivariate multiplicity codes (Kopparty, Saraf, and Yekhanin, J. ACM 2014) are linear codes where the codewords are described by evaluations of multivariate polynomials (with a degree bound) and their derivatives up to a fixed order, on a…
Huffman Codes are optimal Instantaneous Fixed-to-Variable (FV) codes in which every source symbol can only be encoded by one codeword. Relaxing these constraints permits constructing better FV codes. More specifically, recent work has shown…