arXiv subjects

Jamolidin K. Abdurakhmanov

Publications and source records attributed to Jamolidin K. Abdurakhmanov.

3 recordsLinked to original sources

arXiv2026

On the existence of heavy columns in binary matrices with distinct rows

We investigate the existence of heavy columns in binary matrices with distinct rows. A column of an m x n binary matrix is called heavy if the number of ones in it is at least m/2. We introduce two recursive algorithms, A1 and A2, that examine properties of subma trices obtained by row filtering and column deletion. We prove that if algorithm A1 returns True for a binary matrix with distinct rows, then the matrix contains at least one heavy column (Theorem 1). Further more, we prove that if algorithm A2 returns True for a binary matrix with distinct rows, distinct columns, and no all-zero columns, then the matrix also contains at least one heavy column (Theorem 2). The key innovation in A2 is an early termination condition: if exactly one row has a zero in some column, that column is immediately identified as heavy. The proofs employ a novel argument based on the existence of unpaired rows with respect to specific columns, combined with careful analysis of the recursive structure of the algorithms.

cs.DM↗

arXiv2025

Distribution of codewords on the faces of a hypercube and new combinatorial identities

We present a novel framework for studying combinatorial identities through the geometric lens of subset distributions in q-valued cubes. By analyzing how elements of arbitrary subsets are distributed among the faces of the cube E_q^n, we discover new combinatorial identities with geometric significance. We prove that for any subset A contained in E_2^n, the rank function satisfies refined bounds that lead to exact computations for small cardinalities. Specifically, we show that for odd cardinalities, the lower bound is 4D_A/(|A|^2-1) where D_A is the sum of all pairwise Hamming distances in A. Our main theorem establishes identities connecting the number of k-dimensional faces containing exactly e elements of a subset to binomial sums over all subsets of specified cardinality. This yields a parametric family of identities where classical results emerge as special cases. As applications, we derive a geometric interpretation of Vandermonde's identity by examining faces of q-valued cubes, revealing that this classical result naturally arises from counting element distributions. We also obtain a completely new identity for even-weight vectors: (2^(k-1) - 1) times 2^(n-1) times binomial(n,k) equals the sum over i from 1 to floor(n/2) of binomial(n,2i) times binomial(n-2i,k-2i). This identity, valid for all 1 <= k <= n, demonstrates how geometric perspectives can uncover hidden combinatorial relationships. Our framework provides a unified approach for generating new identities and understanding existing ones through subset rank analysis.

cs.DM↗

arXiv2025

The Combinatorial Rank of Subsets: Metric Density in Finite Hamming Spaces

We introduce a novel concept of rank for subsets of finite metric spaces E^n_q (the set of all n-dimensional vectors over an alphabet of size q) equipped with the Hamming distance, where the rank R(A) of a subset A is defined as the number of non-constant columns in the matrix formed by the vectors of A. This purely combinatorial definition provides a new perspective on the structure of finite metric spaces, distinct from traditional linear-algebraic notions of rank. We establish tight bounds for R(A) in terms of D_A, the sum of Hamming distances between all pairs of elements in A. Specifically, we prove that 2qD_A/((q-1)|A|^2) <= R(A) <= D_A/(|A|-1) when |A|/q >= 1, with a modified lower bound for the case |A|/q < 1. These bounds show that the rank is constrained by the metric properties of the subset. Furthermore, we introduce the concept of metrically dense subsets, which are subsets that minimize rank among all isometric images. This notion captures an extremal property of subsets that represent their distance structure in the most compact way possible. We prove that subsets with uniform column distribution are metrically dense, and as a special case, establish that when q is a prime power, every linear subspace of E^n_q is metrically dense. This reveals a fundamental connection between the algebraic and metric structures of these spaces.

cs.DM↗