arXiv subjects

Trygve Johnsen

Publications and source records attributed to Trygve Johnsen.

At least 19 recordsLinked to original sources

arXiv2026

Towards the characterization of minimum weight codewords of Schubert codes

A conjectural formula for the minimum weight of Schubert codes was conjectured by Ghorpade in 2000. This was established by Xiang in 2008. In 2018, Ghorpade and Singh provided a new proof of this conjecture. Moreover, they also conjectured that the minimum weight codewords of the Schubert codes $C_\a (\ell, m)$ are given by the so-called \emph{Schubert decomposable codewords}. We prove the validity of the conjecture for all Schubert varieties $\Omega_{\a}(\ell, V_m)$ for all but finitely many values of $q$.

math.CO↗

arXiv2025

On the hull-variation problem of equivalent vector rank metric codes

The intersection of a linear code with its dual is called the hull of the code. It is known that, for classical linear codes under the Hamming-metric, the dimension of the hull can be reduced up to equivalence. This phenomenon leads to the so-called hull-variation problem formulated by Hao Chen in 2023. In this paper, we consider the analogous problem for vector rank-metric codes, along with their associated matrix codes and extended block codes. Our results include the fact that every vector rank-metric code over any finite field $\mathbb{F}_q$, in particular when $q=2$ or $q=3$, is equivalent to an LCD code.

cs.IT↗

arXiv2025

A q-analogue of delta-matroids and related concepts

We define and study q-delta-matroids, and q-g-matroids. These objects are analogues, for finite-dimensional vector spaces over finite fields, of delta-matroids and g-matroids arising from finite sets. We compare axiomatic descriptions with definitions by means of strong maps of q-matroids.

math.CO↗

arXiv2024

Linear codes associated to symmetric determinantal varieties; General case

The study of linear codes over a finite field of odd cardinality, derived from determinantal varieties obtained from symmetric matrices of bounded rank, was initiated in a recent paper by the authors. There, one found the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank, which is, at most, a given even number. Furthermore, a conjecture for the minimum distance of codes from symmetric matrices with ranks bounded by an odd number was given. In this article, we continue the study of codes from symmetric matrices of bounded rank. A connection between the weights of the codewords of this code and Q-numbers of the association scheme of symmetric matrices is established. Consequently, we get a concrete formula for the weight distribution of these codes. Finally, we determine the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank at most a given number, both when this number is odd and when it is even.

math.AG↗

arXiv2024

Betti Numbers and Higher Weight Spectra of Reed-Muller Codes $RM_q(2,2)$

We determine all the Betti numbers of the $q$-ary second order Reed-Muller codes of length $q^2$, and also of the elongations of matroids associated to these codes. We then use it to determine the higher weight spectra of these codes. As a special case, we recover some results of Kaplan and Matei about counting certain curves over finite fields with prescribed rational intersection points. In geometric terms, our results relate to the affine Veronesean by which we mean the image of the affine plane $\mathbb{A}^2$ under the quadratic Veronese embedding of $\mathbb{P}^2$ in $\mathbb{P}^5$. Indeed, finding the higher weight spectra, of the Reed-Muller code considered here, corresponds to determining the number of $\mathbb{F}_q$-rational points on all possible sections of this affine Veronesean by linear subvarieties of $\mathbb{P}^5$.

math.CO↗

arXiv2023

Linear Codes Associated to Symmetric Determinantal Varieties: Even Rank Case

We consider linear codes over a finite field of odd characteristic, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a code word is derived. Using this formula, we have computed the minimum distance for the codes corresponding to matrices upper-bounded by any fixed, even rank. A conjecture is proposed for the cases where the upper bound is odd. At the end of the article, tables for the weights of these codes, for spaces of symmetric matrices up to order $5$, are given. We also correct typographical errors in Proposition 1.1/3.1 of [3], and in the last table, and we have rewritten Corollary 4.9 of that paper, and the usage of that Corollary in the proof of Proposition 4.10.

cs.IT↗

arXiv2022

Codes from symmetric polynomials

We define and study a class of Reed-Muller type error-correcting codes obtained from elementary symmetric functions in finitely many variables. We determine the code parameters and higher weight spectra in the simplest cases.

cs.IT↗

arXiv2022

The Euler characteristic, $q$-matroids, and a Möbius function

We first give two new proofs of an old result that the reduced Euler characteristic of a matroid complex is equal to the Möbius number of the lattice of cycles of the matroid up to the sign. The purpose has been to find a model to establish an analogous result for the case of $q$-matroids and we find a relation between the Euler characteristic of the simplicial chain complex associated to a $q$-matroid complex and the lattice of $q$-cycles of the $q$-matroid. We use this formula to find the complete homology over $\mathbb{Z}$ of this shellable simplicial complex. We give a characterization of nonzero Euler characteristic for such order complexes. Finally, based on these results we remark why singular homology of a $q$-matroid equipped with order topology may not be effective to describe the $q$-cycles unlike the classical case of matroids.

math.CO↗

arXiv2021

Weight Spectra of Gabidulin Rank-metric Codes and Betti Numbers

We consider $q$-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid associated to the $q$-matroid corresponding to the code. In our main result, we show how these Betti numbers and their elongations determine the generalized weight polynomials for $q$-matorids, in particular, for the Gabidulin rank-metric codes. In addition, we demonstrate how the weight distribution and higher weight spectra of such codes can be determined directly from the associated $q$-matroids by using Möbius functions of its lattice of $q$-flats.

math.CO↗

arXiv2020

Greedy weights for matroids

We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are associated to linear codes, our definitions coincide with those for codes. Thus our Wei duality is a generalization of that for linear codes given by Schaathun. In the last part of the paper we show how some important chains of cycles of the matroids appearing, correspond to chains of component maps of minimal resolutions of the independence complex of the corresponding matroids. We also relate properties of these resolutions to chainedness and greedy weights of the matroids, and in many cases codes, that appear.

cs.IT↗

arXiv2019

A Polymatroid Approach to Generalized Weights of Rank Metric Codes

We consider the notion of a $(q,m)$-polymatroid, due to Shiromoto, and the more general notion of $(q,m)$-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martinez-Penas and Matsumoto for relative generalized rank weights are derived as a consequence.

cs.IT↗

arXiv2019

Higher weight spectra of Veronese codes

We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of the field with q elements. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C

math.CO↗

arXiv2017

Flags of almost affine codes

We describe a two-party wire-tap channel of type II in the framework of almost affine codes. Its cryptological performance is related to some relative profiles of a pair of almost affine codes. These profiles are analogues of relative generalized Hamming weights in the linear case.

cs.IT↗

arXiv2017

Generalized Hamming weights for almost affine codes

We define generalized Hamming weights for almost affine codes. We show how various aspects and applications of generalized Hamming weights for linear codes, such as Wei duality, generalized Kung's bound, profiles, connection to wire-tap channels of type II, apply to the larger class of almost affine codes in general. In addition we discuss duality of almost affine codes,and of the smaller class of multilinear codes. We also give results about weight distributions of infinite series of almost affine codes, each series obtained from a fixed code by extending the code alphabet.

cs.IT↗

arXiv2015

A generalization of weight polynomials to matroids

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid $M$. Our main result is that these polynomials are determined by Betti numbers associated with graded minimal free resolutions of the Stanley-Reisner ideals of $M$ and so-called elongations of $M$. Generalizing Greene's theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.

math.CO↗

arXiv2015

A generalization of Kung's theorem

We give a generalization of Kung's theorem on critical exponents of linear codes over a finite field, in terms of sums of extended weight polynomials of linear codes. For all i=k+1,...,n, we give an upper bound on the smallest integer m such that there exist m codewords whose union of supports has cardinality at least i.

cs.IT↗

arXiv2015

Betti numbers associated to the facet ideal of a matroid

To a matroid M with n edges, we associate the so-called facet ideal F(M) generated by monomials corresponding to bases of M. We show that the Betti numbers related to an N-graded minimal free resolution of F(M) are determined by the Betti numbers related to the blocks of M. Similarly, we show that the higher weight hierarchy of M is determined by the weight hierarchies of the blocks, as well. Drawing on these results, we show that when M is the cycle matroid of a cactus graph, the Betti numbers determine the higher weight hierarchy -- and vice versa. Finally, we demonstrate by way of counterexamples that this fails to hold for outerplanar graphs in general.

math.CO↗

arXiv2013

Stanley-Reisner resolution of constant weight linear codes

Given a constant weight linear code, we investigate its weight hierarchy and the Stanley-Reisner resolution of its associated matroid regarded as a simplicial complex. We also exhibit conditions on the higher weights sufficient to conclude that the code is of constant weight

cs.DM↗