Related papers: Binary (k,k)-designs
This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further…
We apply Schrijver's semidefinite programming method to obtain improved upper bounds on generalized distances and list decoding radii of binary codes.
In this paper we show the usability of the Gray code with constant weight words for computing linear combinations of codewords. This can lead to a big improvement of the computation time for finding the minimum distance of a code. We have…
We obtain new linear programming (LP) and constructive bounds for the covering radius of binary orthogonal arrays of strength $2k$. Our LP bounds develop in two alternative scenarios. First, if a point $y \in F_2^n$, where the covering…
We give one more proof of the first linear programming bound for binary codes, following the line of work initiated by Friedman and Tillich. The new argument is somewhat similar to previous proofs, but we believe it to be both simpler and…
We investigate universal bounds on spherical codes and spherical designs that could be obtained using Delsarte's linear programming methods. We give a lower estimate for the LP upper bound on codes, and an upper estimate for the LP lower…
Alphabetic codes and binary search trees are combinatorial structures that abstract search procedures in ordered sets endowed with probability distributions. In this paper, we design new linear-time algorithms to construct alphabetic codes,…
The dimension of a linear space is the maximum positive integer $d$ such that any $d$ of its points generate a proper subspace. For a set $K$ of integers at least two, recall that a pairwise balanced design PBD$(v,K)$ is a linear space on…
In this paper, we mainly investigate profound interconnections between combinatorial designs, linear codes, and Boolean functions.
We introduce a linear programming method to obtain bounds on the cardinality of codes in Grassmannian spaces for the chordal distance. We obtain explicit bounds, and an asymptotic bound that improves on the Hamming bound. Our approach…
We introduce a new type of $n$-dimensional generalization of symmetric $(v,k,\lambda)$ block designs. We prove upper bounds on the dimension $n$ in terms of $v$ and $k$. We also define the corresponding concept of $n$-dimensional difference…
We derive a linear programming bound on the maximum cardinality of error-correcting codes in the sum-rank metric. Based on computational experiments on relatively small instances, we observe that the obtained bounds outperform all…
Orthogonal array and a large set of orthogonal arrays are important research objects in combinatorial design theory, and they are widely applied to statistics, computer science, coding theory and cryptography. In this paper, some new series…
Unitary t-designs are distributions on the unitary group whose first t moments appear maximally random. Previous work has established several upper bounds on the depths at which certain specific random quantum circuit ensembles approximate…
We study the problem of constructing strong approximate unitary $k$-designs on $D$-dimensional grids (and more generally on Cartesian products of graphs), building on the work of Schuster et al. arXiv:2509.26310 which establishes strong…
New lower bounds on the minimum average Hamming distance of binary codes are derived. The bounds are obtained using linear programming approach.
For large classes of group testing problems, we derive lower bounds for the probability that all significant items are uniquely identified using specially constructed random designs. These bounds allow us to optimize parameters of the…
The Cage Problem requires for a given pair $k \geq 3, g \geq 3$ of integers the determination of the order of a smallest $k$-regular graph of girth $g$. We address a more general version of this problem and look for the $(k,g)$-spectrum of…
We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.
We develop three approaches of combinatorial flavour to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first approach uses techniques from algebraic…