Related papers: Quadratic forms, generalized Hamming weights of co…
Following our joint work arXiv:1003.4578 with Robert Langlands, we make the first steps toward developing geometric methods for analyzing trace formulas in the case of the function field of a curve defined over a finite field. We also…
The interaction of various algebraic structures describing fusion, braiding and group symmetries in quantum projective field theory is an object of an investigation in the paper. Structures of projective Zamolodchikov al- gebras, their…
Polar codes can be viewed as decreasing monomial codes, revealing a rich algebraic structure governed by the lower-triangular affine (LTA) group. We develop a general framework to compute the Hamming weight of codewords generated by sums of…
We provide a comprehensive overview of the fundamental structural properties of weighted projective Reed-Muller codes. We give a recursive construction for these codes, under some conditions for the weights, and we use it to derive bounds…
We classify completely the intersections of the Hermitian curve with parabolas in the affine plane. To obtain our results we employ well-known algebraic methods for finite fields and geometric properties of the curve automorphisms. In…
Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases. In this paper we solve one more special case. The problem of finding the weight distribution is transformed into a…
Quadratic surfaces gain more and more attention among the Geometric Algebra community and some frameworks were proposed in order to represent, transform, and intersect these quadratic surfaces. As far as the authors know, none of these…
In this paper we study the algebraic-geometry of any one-point code on the Hermitian curve. Moreover, we characterize the minimum-weight codewords of some of their dual codes and describe many their small-weight codewords.
We use known characterizations of generalized Paley graphs which are cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the…
Algebraic-geometric codes can be constructed by evaluating a certain set of functions on a set of distinct rational points of an algebraic curve. The set of functions that are evaluated is the linear space of a given divisor or,…
The structure of weighting coefficient matrices of Harmonic Differential Quadrature (HDQ) is found to be either centrosymmetric or skew centrosymmetric depending on the order of the corresponding derivatives. The properties of both matrices…
In the past few years, linear codes with few weights and their weight analysis have been widely studied. In this paper, we further investigate a class of two-weight or three-weight linear codes from defining sets and determine their weight…
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…
In this work we present an explicit relation between the number of points on a family of algebraic curves over $\F_{q}$ and sums of values of certain hypergeometric functions over $\F_{q}$. Moreover, we show that these hypergeometric…
We propose a unified theory of generalized weights for linear codes endowed with an arbitrary distance. Instead of relying on supports or anticodes, the weights of a code are defined via the intersections of the code with a chosen family of…
Coset constructions of $q$-ary Reed-Muller codes can be used to store secrets on a distributed storage system in such a way that only parties with access to a large part of the system can obtain information while still allowing for local…
Using a quadratic version of the Bott residue theorem, we give a quadratic refinement of the count of twisted cubic curves on hypersurfaces and complete intersections in a projective space.
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power $q$, we construct some class of linear code over finite field $\mathbb{F}_q$ with defining set be the preimage of…
Motivated by an application to database linear querying, such as private information-retrieval protocols, we suggest a fundamental property of linear codes -- the generalized covering radius. The generalized covering-radius hierarchy of a…
A method of constructing algebraic-geometric codes with many automorphisms arising from Galois points for algebraic curves is presented.