Related papers: On products and powers of linear codes under compo…

In past few decades, tensor algebra also known as multi-linear algebra has been developed and customized as a tool to be used for various engineering applications. In particular, with the help of a special form of tensor contracted product,…

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…

A fundamental result by L. Solomon in algebraic combinatorics and representation theory states that Mackey formulas for products of characters of a symmetric group, or equivalently the computation of tensor products of representations…

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…

A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the…

We introduce the notion of quadratic hull of a linear code, and give some of its properties. We then show that any symmetric bilinear multiplication algorithm for a finite-dimensional algebra over a field can be obtained by…

Detectability of failures of linear programming (LP) decoding and its potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the LP problem. In this paper, we make a…

This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the…

We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there…

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of…

To solve hard problems, AI relies on a variety of disciplines such as logic, probabilistic reasoning, machine learning and mathematical programming. Although it is widely accepted that solving real-world problems requires an integration…

In this paper we consider the classical problem of computing linear extensions of a given poset which is well known to be a difficult problem. However, in our setting the elements of the poset are multivariate polynomials, and only a small…

In this text we develop some aspects of Harder-Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder-Narasimhan structure associated to a Galois…

Coded computing is a distributed paradigm that uses coding theory to introduce \textit{redundancy} and overcome bottlenecks in large-scale systems. In the same vein, randomized numerical linear algebra employs probabilistic methods to…

We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.

We show that the tensor product of two random linear codes is robustly testable with high probability. This implies that one can obtain pairs of linear codes such that their product and the product of their dual codes are simultaneously…

These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…

We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in…

We propose a graphical language that accommodates two monoidal structures: a multiplicative one for pairing and an additional one for branching. In this colored PROP, whether wires in parallel are linked through the multiplicative structure…

Inspired by prior work by Tian and by Cao and Xu, this paper presents an efficient computer-aided framework to characterize the fundamental limits of coded caching systems under the constraint of linear coding. The proposed framework…