Related papers: Function-Based Minimal Linear Codes over Galois Ri…
Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in \cite{D} constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar…
This paper provides a realization of all classical and most exceptional finite groups of Lie type as Galois groups over function fields over F_q and derives explicit additive polynomials for the extensions. Our unified approach is based on…
Let $p$ be a prime number. Irreducible cyclic codes of length $p^2-1$ and dimension $2$ over the integers modulo $p^h$ are shown to have exactly two nonzero Hamming weights. The construction uses the Galois ring of characteristic $p^h$ and…
In this paper, we investigate hypersurfaces defined over a ring of algebraic integers, and show that if the projection from a point induces a Galois extension over either a number field or the residue field associated with a prime ideal…
Algebraic codes that achieve list decoding capacity were recently constructed by a careful ``folding'' of the Reed-Solomon code. The ``low-degree'' nature of this folding operation was crucial to the list decoding algorithm. We show how…
Coding Theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. Such codes over rings had important applications and many interesting mathematical…
We develop Kummer theory for algebraic function fields in finitely many transcendental variables. We consider any finitely generated Kummer extension (possibly, over a cyclotomic extension) of an algebraic function field, and describe the…
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then…
We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose $q$-expansions satisfy \[ f_k(A, \tau) \colon = q^{-k}(1+a(1)q+a(2)q^2+...) + O(q),\] where $a(n)$ are…
In this short survey we concern ourselves with minimal codes, a classical object in coding theory. We will explain the relation between minimal codes and various other mathematical domains, in particular with finite projective geometry.…
In their fundamental paper published in 1965, G. Solomon and J. J. Stiffler invented infinite families of codes meeting the Griesmer bound. These codes are then called Solomon-Stiffler codes and have motivated various constructions of codes…
Linear codes are the most important family of codes in cryptography and coding theory. Some codes have only a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs.…
Using the connection with the Frobenius manifold structure, we study the matrix model description of minimal Liouville gravity (MLG) based on the Douglas string equation. Our goal is to find an exact discrete formulation of the (q,p) MLG…
Galois self-orthogonal (SO) codes are generalizations of Euclidean and Hermitian SO codes. Algebraic geometry (AG) codes are the first known class of linear codes exceeding the Gilbert-Varshamov bound. Both of them have attracted much…
The results of [1,2] on linear homogeneous two-weight codes over finite Frobenius rings are exended in two ways: It is shown that certain non-projective two-weight codes give rise to strongly regular graphs in the way described in [1,2].…
We construct new linear codes with high minimum distance d. In at least 12 cases these codes improve the minimum distance of the previously known best linear codes for fixed parameters n,k. Among these new codes there is an optimal ternary…
Blocking sets and minimal codes have been studied for many years in projective geometry and coding theory. In this paper, we provide a new lower bound on the size of $t$-fold $s$-blocking sets without the condition $t \leq q$, which is…
In this work, we study codes over the ring R_{k,m}=F_2[u,v]/<u^{k},v^{m},uv-vu>, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from R_{k,m} to…
We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the…
In coding theory, constructing codes with good parameters is one of the most important and fundamental problems. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers.…