相关论文: Minimal linear codes in odd characteristic
In two previous papers we constructed new families of completely regular codes by concatenation methods. Here we determine cases in which the new codes are completely transitive. For these cases we also find the automorphism groups of such…
In [10], the existence of $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$, with dimension $12$, minimum distance $5$ and left idealiser isomorphic to $\mathbb{F}_{q^6}$, defined by a trinomial of $\mathbb{F}_{q^6}[x]$, when…
We construct MDS Euclidean and Hermitian self-dual codes over large finite fields of odd and even characteristics. Our codes arise from cyclic and negacyclic duadic codes.
Using a general result of Lusztig, we find the decomposition into irreducibles of certain induced characters of the projective general linear group over a finite field of odd characteristic.
The order bound for the minimum distance of algebraic geometry codes was originally defined for the duals of one-point codes and later generalized for arbitrary algebraic geometry codes. Another bound of order type for the minimum distance…
Let $A_q(n,d)$ be the maximum order (maximum number of codewords) of a $q$-ary code of length $n$ and Hamming distance at least $d$. And let $A(n,d,w)$ that of a binary code of constant weight $w$. Building on results from algebraic graph…
R. W. Hamming published the Hamming codes and the sphere packing bound in 1950. In the past 75 years, infinite families of distance-optimal linear codes over finite fields with minimum distance at most 8 with respect to the sphere packing…
We present a new bound for the minimum distance of a general primary linear code. For affine variety codes defined from generalised C_{ab} curves the new bound often improves dramatically on the Feng-Rao bound for primary codes. The method…
For various nonsolvable groups $G$, we prove the existence of extensions of the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set for $G$.…
The task of constructing infinite families of self-dual codes with unbounded lengths and minimum distances exhibiting square-root lower bounds is extremely challenging, especially when it comes to cyclic codes. Recently, the first infinite…
In this paper, we investigate the existence of self-dual MRD codes $C\subset L^n$, where $L/F$ is an arbitrary field extension of degree $m\geq n$. We then apply our results to the case of finite fields, and prove that if $m=n$ and…
We present a new general construction of MDS codes over a finite field $\mathbb{F}_q$. We describe two explicit subclasses which contain new MDS codes of length at least $q/2$ for all values of $q \ge 11$. Moreover, we show that most of the…
Linear codes with few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of…
Let $q$ be a prime power. This paper provides a new class of linear codes that arises from the action of the alternating group on $\mathbb F_q[x_1,\dots,x_m]$ combined with the ideas in (M. Datta and T. Johnsen, 2022). Compared with…
By solving an infinite nonlinear system of $q$-difference equations one constructs a chain of $q$-difference operators. The eigenproblems for the chain are solved and some applications, including the one related to $q$-Hahn orthogonal…
Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic.…
Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in…
Some linear codes associated to maximal algebraic curves via Feng-Rao construction are investigated. In several case, these codes have better minimum distance with respect to the previously known linear codes with same length and dimension.
We show that many classical results of the minimal model programme do not hold over an algebraically closed field of characteristic two. Indeed, we construct a three dimensional plt pair whose codimension one part is not normal, a three…
F.-H. Lin studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an…