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The defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields, whenever positive characteristic is involved. We continue the detailed study of defect…
Maximum distance profile codes are characterized by the property that two trajectories which start at the same state and proceed to a different state will have the maximum possible distance from each other relative to any other…
A distance measure is presented between two unitary propagators of quantum systems of differing dimensions along with a corresponding method of computation. A typical application is to compare the propagator of the actual (real) process…
Two families of complementary codes over finite fields $\mathbb{F}_q$ are studied, where $q=r^2$ is square: i) Hermitian complementary dual linear codes, and ii) trace Hermitian complementary dual subfield linear codes. Necessary and…
We ask what is the general framework for a quantum error correcting code that is defined by a sequence of measurements. Recently, there has been much interest in Floquet codes and space-time codes. In this work, we define and study the…
The free distance of a convolutional code is a reliable indicator of its performance. However its computation is not an easy task. In this paper, we present some algorithms to compute the free distance with good efficiency that work for…
A basic problem for the constant dimension subspace coding is to determine the maximal possible size A_q (n, d, k) of a set of k-dimensional subspaces in Fnq such that the subspace distance satisfies d(U, V )> or =d for any two different…
This paper provides new constructions and lower bounds for subspace codes, using Ferrers diagram rank-metric codes from matchings of the complete graph and pending blocks. We present different constructions for constant dimension codes with…
Let $\mathbb{F}_q$ be a finite field of order $q$ and $E$ be a set in $\mathbb{F}_q^d$. The distance set of $E$ is defined by $\Delta(E):=\{\lVert x-y \rVert :x,y\in E\}$, where $\lVert \alpha \rVert=\alpha_1^2+\dots+\alpha_d^2$. Iosevich,…
A vector space partition of $\mathbb{F}_q^v$ is a collection of subspaces such that every non-zero vector is contained in a unique element. We improve a lower bound of Heden, in a subcase, on the number of elements of the smallest occurring…
For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix, and the distance energy is defined as the sum of the absolute values of the eigenvalues of its distance matrix. We establish lower and…
For nonnegative integers $n_2, n_3$ and $d$, let $N(n_2,n_3,d)$ denote the maximum cardinality of a code of length $n_2+n_3$, with $n_2$ binary coordinates and $n_3$ ternary coordinates (in this order) and with minimum distance at least…
In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are…
Let $q,d\geq 2$ be integers. Define $$ J(q,d):=\frac 1q \Big( \min_{0<x<1} \frac{1-x^q}{1-x} x^{-\frac{q-1}{d}}\Big). $$ Let $\mbox{$\cal G$}\subseteq {\mathbb R}^n$ be an arbitrary subset. We denote by $d(\mbox{$\cal G$})$ the set of…
A spherical two-distance set is a finite collection of unit vectors in $\reals^n$ such that the set of distances between any two distinct vectors has cardinality two. We use the semidefinite programming method to compute improved estimates…
The {\em bottleneck distance} is a natural measure of the distance between two finite point sets of equal cardinality, defined as the minimum over all bijections between the point sets of the maximum distance between any pair of points put…
Equidistant codes over vector spaces are considered. For $k$-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest…
q-ary cumulative-separable $\Gamma(L,G^{(j)})$-codes $L=\{ \alpha \in GF(q^{m}):G(\alpha )\neq 0 \}$ and $G^{(j)}(x)=G(x)^{j}, 1 \leq i\leq q$ are considered. The relation between different codes from this class is demonstrated. Improved…
Self-orthogonal $J$-affine variety codes have been successfully used to obtain quantum stabilizer codes with excellent parameters. In a previous paper we gave formulae for the dimension of this family of quantum codes, but no bound for the…
A Locally Recoverable Code is a code such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. When we have $\delta$ non overlapping subsets of cardinality $r_i$ that…