Trygve Johnsen
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View article: On the hull-variation problem of equivalent vector rank metric codes
On the hull-variation problem of equivalent vector rank metric codes Open
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…
View article: Linear codes associated to symmetric determinantal varieties; General case
Linear codes associated to symmetric determinantal varieties; General case Open
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 …
View article: Higher weight spectra and Betti numbers of Reed-Muller codes $RM_q(2,2)$
Higher weight spectra and Betti numbers of Reed-Muller codes $RM_q(2,2)$ Open
We determine the higher weight spectra of $q$-ary Reed-Muller codes $C_q=RM_q(2,2)$ for all prime powers $q$. This is equivalent to finding the usual weight distributions of all extension codes of $C_q$ over every field extension of $F_q$ …
View article: A q-analogue of delta-matroids and related concepts
A q-analogue of delta-matroids and related concepts Open
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…
View article: The Euler characteristic, $q$-matroids, and a Möbius function
The Euler characteristic, $q$-matroids, and a Möbius function Open
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 establis…
View article: Weight spectra of Gabidulin rank-metric codes and Betti numbers
Weight spectra of Gabidulin rank-metric codes and Betti numbers Open
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 assoc…
View article: Codes from symmetric polynomials
Codes from symmetric polynomials Open
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.
View article: Möbius and coboundary polynomials for matroids
Möbius and coboundary polynomials for matroids Open
We study how some coefficients of two-variable coboundary polynomials can be derived from Betti numbers of Stanley–Reisner rings. We also explain how the connection with these Stanley–Reisner rings forces the coefficients of the two-variab…
View article: Weight Spectra of Gabidulin Rank-metric Codes and Betti Numbers
Weight Spectra of Gabidulin Rank-metric Codes and Betti Numbers Open
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 asso…
View article: Linear Codes Associated to Symmetric Determinantal Varieties: Even Rank Case
Linear Codes Associated to Symmetric Determinantal Varieties: Even Rank Case Open
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 co…
View article: Higher Weight Spectra of Veronese Codes
Higher Weight Spectra of Veronese Codes Open
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 extens…
View article: Flags of almost affine codes
Flags of almost affine codes Open
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 gene…