Related papers: Rank Bounds for Design Matrices with Applications …
Error-correcting codes resilient to synchronization errors such as insertions and deletions are known as insdel codes. Due to their important applications in DNA storage and computational biology, insdel codes have recently become a focal…
A fascinating topic of combinatorics is $t$-designs, which have a very long history. The incidence matrix of a $t$-design generates a linear code over GF$(q)$ for any prime power $q$, which is called the linear code of the $t$-design over…
Recently, it was discovered by several authors that a $q$-ary optimal locally recoverable code, i.e., a locally recoverable code archiving the Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the…
The concept of linear set in projective spaces over finite fields was introduced by Lunardon in 1999 and it plays central roles in the study of blocking sets, semifields, rank-distance codes and etc. A linear set with the largest possible…
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric.…
We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if…
Combinatorial $t$-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $t$-designs has been…
It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a…
We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the…
Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by…
We consider the matrix completion problem with a deterministic pattern of observed entries. In this setting, we aim to answer the question: under what condition there will be (at least locally) unique solution to the matrix completion…
The explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these…
We consider a class of linear codes associated to projective algebraic varieties defined by the vanishing of minors of a fixed size of a generic matrix. It is seen that the resulting code has only a small number of distinct weights. The…
The binary rank of a $0,1$ matrix is the smallest size of a partition of its ones into monochromatic combinatorial rectangles. A matrix $M$ is called $(k_1, \ldots, k_m ; n_1, \ldots, n_m)$ circulant block diagonal if it is a block matrix…
The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection between this characterization and polarities of projective geometries is explained.…
The $2$-to-$1$ mapping over finite fields has a wide range of applications, including combinatorial mathematics and coding theory. Thus, constructions of $2$-to-$1$ mappings have attracted considerable attention recently. Based on…
Klee's measure problem (computing the volume of the union of $n$ axis-parallel boxes in $\mathbb{R}^d$) is well known to have $n^{\frac{d}{2}\pm o(1)}$-time algorithms (Overmars, Yap, SICOMP'91; Chan FOCS'13). Only recently, a conditional…
In the recent work \cite{shi18}, a combinatorial problem concerning linear codes over a finite field $\F_q$ was introduced. In that work the authors studied the weight set of an $[n,k]_q$ linear code, that is the set of non-zero distinct…
The performance of Reed--Solomon codes (RS codes, for short) in the presence of insertion and deletion errors has attracted growing attention in recent literature. In this work, we further study this intriguing mathematical problem,…
Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An…