Related papers: Polytope Representations for Linear-Programming De…
There are two gradient descent decoding procedures for binary codes proposed independently by Liebler and by Ashikhmin and Barg. Liebler in his paper mentions that both algorithms have the same philosophy but in fact they are rather…
Code loops are Moufang loops constructed from doubly even binary codes. Then, given a code loop L, we ask which doubly even binary code V produces L. In this sense, V is called a representation of L. In this article we define and determine…
In this paper, we consider quantized decoding of LDPC codes on the binary symmetric channel. The binary message passing algorithms, while allowing extremely fast hardware implementation, are not very attractive from the perspective of…
Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer…
Decoding a Reed-Solomon code can be modeled by a bilinear system which can be solved by Gr\"obner basis techniques. We will show that in this particular case, these techniques are much more efficient than for generic bilinear systems with…
Numerical experiments are performed in order to study the performance of ABS codes in solving non-linear systems of equations.
Matrix (or operator) recovery from linear measurements is a well-studied problem. However, there are situations where only bilinear or quadratic measurements are available. A bilinear or quadratic problem can easily be transformed into a…
Recently, linear codes constructed by defining sets have attracted a lot of study, and many optimal linear codes with a few weights have been produced. The objective of this paper is to present a class of binary linear codes with three…
The constrained linear representability problem (CLRP) for polymatroids determines whether there exists a polymatroid that is linear over a specified field while satisfying a collection of constraints on the rank function. Using a computer…
A new permutation decoding approach for polar codes is presented. The complexity of the algorithm is similar to that of a successive cancellation list (SCL) decoder, while it can be implemented with the latency of a successive cancellation…
Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary…
Decoding error-correctiong codes by methods of mathematical optimization, most importantly linear programming, has become an important alternative approach to both algebraic and iterative decoding methods since its introduction by Feldman…
Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is…
We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace…
This work is on fast encoding and decoding of polar codes. We propose and detail 8-bit and 16-bit parallel decoders that can be used to reduce the decoding latency of the successive-cancellation decoder. These decoders are universal and can…
We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…
A binary linear code $C$ is a $\mathbb{Z}_2$-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be…
The design of codes for feedback-enabled communications has been a long-standing open problem. Recent research on non-linear, deep learning-based coding schemes have demonstrated significant improvements in communication reliability over…
We study fountain codes transmitted over the binary-input symmetric-output channel. For channels with small capacity, receivers needs to collects many channel outputs to recover information bits. Since a collected channel output yields a…
We present an algorithm for the classification of linear codes over finite fields, based on lattice point enumeration. We validate a correct implementation of our algorithm with known classification results from the literature, which we…