Related papers: Maximal arcs, codes, and new links between project…
A maximal arc of degree k in a finite projective plane P of order q = ks is a set of (q-s+1)k points that meets every line of P in either k or 0 points. The collection of the nonempty intersections of a maximal arc with the lines of P is a…
In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in [C. Ding, C. Li, Infinite families of 2-designs and 3-designs from linear codes,…
All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact…
It is known that some good linear codes over a finite ring (R-linear codes) arise from interesting point constellations in certain projective geometries. For example, the expurgated Nordstrom-Robinson code, a nonlinear binary [14,6,6]-code…
Subspace codes are collections of subspaces of a projective space such that any two subspaces satisfy a pairwise minimum distance criterion. Recent results have shown that it is possible to construct optimal $(5,3)$ subspace codes from…
The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising frommaximal (120,8)-arcs in the known projective planes of order 16 are computed. It is shown that each of these designs is embeddable in a unique…
Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for…
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear…
New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for q <= 9109. From these new bounds it follows that for q <= 2621 and q = 2659,2663,2683,2693,2753,2801, the relation…
It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$…
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking…
Codes defined on graphs and their properties have been subjects of intense recent research. On the practical side, constructions for capacity-approaching codes are graphical. On the theoretical side, codes on graphs provide several…
In this paper we will estimate the main parameters of some evaluation codes which are known as projective parameterized codes. We will find the length of these codes and we will give a formula for the dimension in terms of the Hilbert…
The binary Hamming codes with parameters $[2^m-1, 2^m-1-m, 3]$ are perfect. Their extended codes have parameters $[2^m, 2^m-1-m, 4]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes…
An (n,r)-arc in PG(2,q) is a set of n points such that each line contains at most r of the selected points. It is well-known that (n,r)-arcs in PG(2,q) correspond to projective linear codes. Let m_r(2,q) denote the maximal number n of…
A combinatorial characterization of resolvable Steiner 2-$(v,k,1)$ designs embeddable as maximal arcs in a projective plane of order $(v-k)/(k-1)$ is proved, and a generalization of a conjecture by Andries Brouwer \cite{Br} is formulated.
In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and…
The $p$-ary code associated with the incidence structure of points and $t$-spaces in a projective space $\mathrm{PG}(m,q)$, where $q=p^h$, is the $\mathbb{F}_p$-subspace generated by the incidence vectors of the blocks of this design. The…
Linear codes in the projective space $\mathbb{P}_q(n)$, the set of all subspaces of the vector space $\mathbb{F}_q^n$, were first considered by Braun, Etzion and Vardy. The Grassmannian $\mathbb{G}_q(n,k)$ is the collection of all subspaces…
The maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ is $257$, where the $2$ isomorphism types are extended lifted maximum rank distance codes. In…