Related papers: Near optimal constructions of frameproof codes
Significant work has been done on computing the ``average'' optimal solution value for various $\mathsf{NP}$-complete problems using the Erd\"{o}s-R\'{e}nyi model to establish \emph{critical thresholds}. Critical thresholds define narrow…
A code over a finite field is called locally recoverable code (LRC) if every coordinate symbol can be determined by a small number (at most r, this parameter is called locality) of other coordinate symbols. For a linear code with length n,…
In this paper, we study flag codes on the vector space $\mathbb{F}_q^n$, being $q$ a prime power and $\mathbb{F}_q$ the finite field of $q$ elements. More precisely, we focus on flag codes that attain the maximum possible distance (optimum…
An $(n,r,h,a,q)$-Local Reconstruction Code (LRC) is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $n/r$ local groups each of size $r$. Each local group satisfies `$a$' local parity checks to…
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…
We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in…
Mixed Hamming packings are considered: the maximal cardinality given a minimum codeword Hamming distance of mixed codes is addressed via mixed integer programming models. Adopting the concept of contact graph from classical continuous…
Locally repairable codes are widely applicable in contemporary large-scale distributed cloud storage systems and various other areas. By making use of some algebraic structures of elliptic curves, Li et al. developed a series of $q$-ary…
Recent work has reemphasized the importance of cardinality estimates for query optimization. While new techniques have continuously improved in accuracy over time, they still generally allow for under-estimates which often lead optimizers…
It is shown that some well-known and some new cyclic codes with orthogonal parity-check equations can be constructed in the finite-field transform domain. It is also shown that, for some binary linear cyclic codes, the performance of the…
The decomposition theory of matroids initiated by Paul Seymour in the 1980's has had an enormous impact on research in matroid theory. This theory, when applied to matrices over the binary field, yields a powerful decomposition theory for…
We generalized to higher dimensions the notions of optical orthogonal codes. We establish uper bounds on the capacity of general $ n $-dimensional OOCs, and on specific types of ideal codes (codes with zero off-peak autocorrelation). The…
We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower…
Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this paper, we give FPT algorithms for the…
In this paper q-ary Raptor codes under ML decoding are considered. An upper bound on the probability of decoding failure is derived using the weight enumerator of the outer code, or its expected weight enumerator if the outer code is drawn…
Algebraic Manipulation Detection (AMD) codes is a cryptographic primitive that was introduced by Cramer, Dodis, Fehr, Padro and Wichs. They are keyless message authentication codes that protect messages against additive tampering by the…
There are several problems in the theory of online computation where tight lower bounds on the competitive ratio are unknown and expected to be difficult to describe in a short form. A good example is the Online Bin Stretching problem, in…
Batch codes are a useful notion of locality for error correcting codes, originally introduced in the context of distributed storage and cryptography. Many constructions of batch codes have been given, but few lower bound (limitation)…
A spherical $L$-code, where $L \subseteq [-1,\infty)$, consists of unit vectors in $\mathbb{R}^d$ whose pairwise inner products are contained in $L$. Determining the maximum cardinality $N_L(d)$ of an $L$-code in $\mathbb{R}^d$ is a…
In this paper, a link between polymatroid theory and locally repairable codes (LRCs) is established. The codes considered here are completely general in that they are subsets of $A^n$, where $A$ is an arbitrary finite set. Three classes of…