Related papers: Decoding Downset codes over a finite grid
In [4] we describe a variation of the classical permutation decoding algorithm that can be applied to any binary affine-invariant code; in particular, it can be applied to first-order Reed-Muller codes successfully. In this paper we study…
Information set decoding (ISD) algorithms are the best known procedures to solve the decoding problem for general linear codes. These algorithms are hence used for codes without a visible structure, or for which efficient decoders…
A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making…
Polynomial evaluation codes hold a prominent place in coding theory. In this work, we study the problem of list decoding for a general class of polynomial evaluation codes, also known as Toric codes, that are defined for any given convex…
The classical family of Reed-Solomon codes consist of evaluations of polynomials over the finite field $\mathbb{F}_q$ of degree less than $k$, at $n$ distinct field elements. These are arguably the most widely used and studied codes, as…
Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, monomials and trinomials over…
Here we present some revised arguments to a randomized algorithm proposed by Sudan to find the polynomials of bounded degree agreeing on a dense fraction of a set of points in $\mathbb{F}^{2}$ for some field $\mathbb{F}$.
A complex spherical code is a finite subset on the unit sphere in $\mathbb{C}^d$. A fundamental problem on complex spherical codes is to find upper bounds for those with prescribed inner products. In this paper, we determine the irreducible…
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…
In this paper, we propose an analysis of the automorphism group of polar codes, with the scope of designing codes tailored for automorphism ensemble (AE) decoding. We prove the equivalence between the notion of decreasing monomial codes and…
In this work we show how to decompose a linear code relatively to any given poset metric. We prove that the complexity of syndrome decoding is determined by a maximal (primary) such decomposition and then show that a refinement of a partial…
Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is to choose a suitable dominating set $C$ of a graph $G$ which is also…
Permutation decoding is a technique which involves finding a subset $S$, called PD-set, of the permutation automorphism group of a code $C$ in order to assist in decoding. An explicit construction of $\left \lfloor{\frac{2^m-m-1}{1+m}}…
A unique decoding algorithm for general AG codes, namely multipoint evaluation codes on algebraic curves, is presented. It is a natural generalization of the previous decoding algorithm which was only for one-point AG codes. As such, it…
Several recently proposed code-based cryptosystems base their security on a slightly generalized version of the classical (syndrome) decoding problem. Namely, in the so-called restricted (syndrome) decoding problem, the error values stem…
An output-polynomial algorithm for the listing of minimal dominating sets in graphs is a challenging open problem and is known to be equivalent to the well-known Transversal problem which asks for an output-polynomial algorithm for listing…
In this work, two algorithms are developed related to lattice codes. In the first one, an extended complete Gr\"obner basis is computed for the label code of a lattice. This basis supports all term orderings associated with a total degree…
Subspace codes have received an increasing interest recently due to their application in error-correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and…
Tensor codes are a generalisation of matrix codes. Such codes are defined as subspaces of order-r tensors for which the ambient space is endowed with the tensor-rank as a metric. A class of these codes was introduced by Roth, who also…
Decoding algorithms are essential to fault-tolerant quantum-computing architectures. In this perspective we explore decoding algorithms for the surface code; a prototypical quantum low-density parity-check code that underlies many of the…