Related papers: A lower bound on permutation codes of distance $n-…
Let $\alpha:\mathbb{Z}_n\to\mathbb{Z}_n$ be a permutation and $\Psi(\alpha)$ be the number of collinear triples modulo $n$ in the graph of $\alpha$. Cooper and Solymosi had given by induction the bound…
MDS convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree. In this paper we introduce a class of MDS convolutional codes whose column distances reach the generalized…
We construct maximally recoverable codes (corresponding to partial MDS codes) which are based on linearized Reed-Solomon codes. The new codes have a smaller field size requirement compared with known constructions. For certain asymptotic…
Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$…
We consider quantum MDS (QMDS) codes for quantum systems of dimension $q$ with lengths up to $q^2+2$ and minimum distances up to $q+1$. We show how starting from QMDS codes of length $q^2+1$ based on cyclic and constacyclic codes, new QMDS…
We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show…
We show there is a natural connection between Latin squares and commutative sets of monomials defining geometric structures in finite phase-space of prime power dimensions. A complete set of such monomials defines a mutually unbiased basis…
By developing one-shot mutual covering lemmas, we derive a one-shot achievability bound for broadcast with a common message which recovers Marton's inner bound (with three auxiliary random variables) in the i.i.d.~case. The encoder employed…
Kim et al. (2021) gave a method to embed a given binary $[n,k]$ code $\mathcal{C}$ $(k = 3, 4)$ into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d' \ge d(\mathcal{C})$. We extend this…
Permutation codes and multi-permutation codes have been widely considered due to their various applications, especially in flash memory. In this paper, we consider permutation codes and multi-permutation codes against a burst of stable…
Let $\mathbb{F}_q$ be a finite field with $q=p^{e}$ elements, where $p$ is a prime number and $e \geq 1$ is an integer. In this paper, by means of generalized Reed-Solomon (GRS) codes, we construct two new classes of quantum…
We examine an error-correcting coding framework in which each coded symbol is constrained to be a function of a fixed subset of the message symbols. With an eye toward distributed storage applications, we seek to design systematic codes…
In this note, we give a construction that provides a tight lower bound of mn-1 for the length of the shortest word in the intersection of two regular languages with state complexities m and n.
We study first-passage percolation in two dimensions, using measures mu on passage times with b:=inf supp(mu) >0 and mu({b})=p \geq p_c, the threshold for oriented percolation. We first show that for each such mu, the boundary of the limit…
Mixed (asymmetric) orthogonal arrays (MOAs) generalize classical orthogonal arrays by allowing columns over different alphabets. However, their study requires very different structural tools than those used for symmetric orthogonal arrays…
Constructions of quantum MDS codes have been studied by many authors. We refer to the table in page 1482 of [3] for known constructions. However there are only few $q$-ary quantum MDS $[[n,n-2d+2,d]]_q$ codes with minimum distances…
The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding…
Two upper bounds on the minimum distance of type-1 quasi-cyclic low-density parity-check (QC LDPC) codes are derived. The necessary condition is given for the minimum code distance of such codes to grow linearly with the code length.
Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are generalized Reed-Solomon (GRS) codes. The square…
Constructions of optimal locally repairable codes (LRCs) in the case of $(r+1) \nmid n$ and over small finite fields were stated as open problems for LRCs in [I. Tamo \emph{et al.}, "Optimal locally repairable codes and connections to…