Related papers: Coding theory package for Macaulay2
Using the geometric vertex decomposition property first defined by Knutson, Miller, and Yong, a recursive definition for geometrically vertex decomposable ideals was given by Klein and Rajchgot. We introduce the Macaulay2 package…
The hull of a linear code is defined as the intersection of the code and its dual. This concept was initially introduced to classify finite projective planes. The hull plays a crucial role in determining the complexity of algorithms used to…
Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. There are several methods to construct linear codes, one of which is based on functions over finite fields. Recently, many…
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful…
Recently, linear codes constructed from defining sets have been studied extensively. They may have nice parameters if the defining set is chosen properly. Let $ m >2$ be a positive integer. For an odd prime $ p $, let $ r=p^m $ and…
We compute the basic parameters (dimension, length, minimum distance) of affine evaluation codes defined on a cartesian product of finite sets. Given a sequence of positive integers, we construct an evaluation code, over a degenerate torus,…
In this paper, we first generalize the class of linear codes by Ding and Ding (IEEE TIT, 61(11), pp. 5835-5842, 2015). Then we mainly study the augmented codes of this generalized class of linear codes. For one thing, we use Gaussian sums…
We introduce the package LatticePolytopes for Macaulay2. The package provides methods for computations related to Cayley structures, local positivity and smoothness for lattice polytopes.
Two broad classes of graphical modeling problems for codes can be identified in the literature: constructive and extractive problems. The former class of problems concern the construction of a graphical model in order to define a new code.…
Large language models (LLMs) have shown remarkable ability to generate code, yet their outputs often violate syntactic or semantic constraints when guided only through natural language prompts. We introduce TreeCoder, the most general and…
We present the Macaulay2 package TropicalToric.m2 for toric intersection theory computations using tropical geometry.
When attempting to understand the behavior of an executable, a binary analyst can make use of many different techniques. These include program slicing, dynamic instrumentation, binary-level rewriting, symbolic execution, and formal…
As a result of their applications in network coding, space-time coding, and coding for criss-cross errors, matrix codes have garnered significant attention; in various contexts, these codes have also been termed rank-metric codes,…
The challenge in learning abstract concepts from images in an unsupervised fashion lies in the required integration of visual perception and generalizable relational reasoning. Moreover, the unsupervised nature of this task makes it…
In this paper, we develop a new decoding algorithm of a binary linear codes for symbol-pair read channels. Symbol-pair read channel has recently been introduced by Cassuto and Blaum to model channels with high write resolution but low read…
Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial $t$-designs have been attracted lots of research interest for decades. The interplay between coding theory and $t$-designs has…
We consider the scenario in which a set of sources generate messages in a network and a receiver node demands an arbitrary linear function of these messages. We formulate an algebraic test to determine whether an arbitrary network can…
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…
Code search is vital in the maintenance and extension of software systems. Past works have used separate language models for the natural language and programming language artifacts on models with multiple encoders and different loss…
The traditional definition of Integrated Interleaved (II) codes generally assumes that the component nested codes are either Reed-Solomon (RS) or shortened Reed-Solomon codes. By taking general classes of codes, we present a recursive…