Related papers: Binary (k,k)-designs
A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in $Q$-polynomial distance-regular…
The linear programming method is applied to the space $\U_n(\C)$ of unitary matrices in order to obtain bounds for codes relative to the diversity sum and the diversity product. Theoretical and numerical results improving previously known…
A combinatorial design is a family of sets that are almost disjoint, which is applied in pseudo random number generations and randomness extractions. The parameter, $\rho$, quantifying the overlap between the sets within the family, is…
In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the…
In recent years, many connections have been made between minimal codes, a classical object in coding theory, and other remarkable structures in finite geometry and combinatorics. One of the main problems related to minimal codes is to give…
We generalize the notion of linear chord diagrams to the case of matched sets of size $k$, which we call $k$-chord diagrams. We provide formal generating functions and recurrence relations enumerating these $k$-chord diagrams by the number…
A binary $[n,k]$-linear code $\mathcal{C}$ is a $k$-dimensional subspace of $\mathbb{F}_2^n$. For $\boldsymbol{x}\in \mathbb{F}_2^n$, the set $\boldsymbol{x}+\mathcal{C}$ is a coset of $\mathcal{C}$. In this work we study a partial ordering…
We construct $\varepsilon$-approximate unitary $k$-designs on $n$ qubits in circuit depth $O(\log k \log \log n k / \varepsilon)$. The depth is exponentially improved over all known results in all three parameters $n$, $k$, $\varepsilon$.…
The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
Let $n=2^k-1$ and $m=2^{k-2}$ for a certain $k\ge 3$. Consider the point-line geometry of $2m$-element subsets of an $n$-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension $k$. For $k\ge…
Let $G$ be a connected graph. For an ordered set $S=\{v_1,\ldots, v_\ell\}\subseteq V(G)$, the vector $r_G(v|S) = (d_G(v_1,v), \ldots, d_G(v_\ell,v))$ is called the metric $S$-representation of $v$. If for any pair of different vertices…
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…
In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients…
We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global…
A unitary 2-design can be viewed as a quantum analogue of a 2-universal hash function: it is indistinguishable from a truly random unitary by any procedure that queries it twice. We show that exact unitary 2-designs on n qubits can be…
Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in simple situations. This paper concerns binary radix systems, i.e., positional number systems with digits…
We consider quantum, nondterministic and probabilistic versions of known computational model Ordered Read-$k$-times Branching Programs or Ordered Binary Decision Diagrams with repeated test ($k$-QOBDD, $k$-NOBDD and $k$-POBDD). We show…
Combinatorial $t$-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a $t$-design. Till now only a small amount of…
In this work, we study two types of constraints on two-dimensional binary arrays. In particular, given $p,\epsilon>0$, we study (i) The $p$-bounded constraint: a binary vector of size $m$ is said to be $p$-bounded if its weight is at most…