Related papers: A Singleton Bound for Lattice Schemes
Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal…
The discrete multiscale analysis for boundary value problems in nonlinear discrete systems leads to a first order discrete modulational instability above a threshold amplitude for wave numbers beyond the zero of group velocity dispersion.…
In this paper, we prove that explicit FRS codes and multiplicity codes achieve relaxed generalized Singleton bounds for list size $L\ge1.$ Specifically, we show the following: (1) FRS code of length $n$ and rate $R$ over the alphabet…
In this work, doubly extended linearized Reed--Solomon codes and triply extended Reed--Solomon codes are generalized. We obtain a general result in which we characterize when a multiply extended code for a general metric attains the…
We are concerned with linear redundancy storage schemes regarding their ability to provide concurrent (local) recovery of multiple data objects. This paper initiates a study of such systems within the classical coding theory. We show how we…
In this work, we determine new linear equations for the weight distribution of linear codes over finite chain rings. The identities are determined by counting the number of some special submatrices of the parity-check matrix of the code.…
Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for…
We study finite-field extensions that preserve the same support as the parity-check matrices defining a given binary CSS code. Here, an LDPC-CSS code refers to a CSS code whose parity-check matrices are orthogonal in the sense that each…
The compass model on a square lattice provides a natural template for building subsystem stabilizer codes. The surface code and the Bacon-Shor code represent two extremes of possible codes depending on how many gauge qubits are fixed. We…
String compactification in the framework of the low energy effective supergravity requires the perturbative control in both the large volume and the weak coupling expansions. However, when the complex structure moduli couple to some lattice…
Self-dual codes have been studied actively because they are connected with mathematical structures including block designs and lattices and have practical applications in quantum error-correcting codes and secret sharing schemes.…
One of the main weakness of the family of centralizer codes is that its length is always $n^2$. Thus we have taken a new matrix equation code called intertwining code. Specialty of this code is the length of it, which is of the form $nk$.…
Linear constraints for a matrix polytope with no fractional vertex are investigated as intersecting research among permutation codes, rank modulations, and linear programming methods. By focusing the discussion to the block structure of…
Network coding is studied when an adversary controls a subset of nodes in the network of limited quantity but unknown location. This problem is shown to be more difficult than when the adversary controls a given number of edges in the…
We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of…
Cyclic codes are an important class of linear codes. Bounding the minimum distance of cyclic codes is a long-standing research topic in coding theory, and several well-known and basic results have been developed on this topic. Recently,…
We present and prove a well-known locality bound for the complex-time dynamics of a general class of one-dimensional quantum spin systems. Then we discuss how one might hope to extend this same procedure to higher dimensions using ideas…
For a many-to-many matching market, we study the lattice structure of the set of random stable matchings. We define a partial order on the random stable set and present two intuitive binary operations to compute the least upper bound and…
The iterative decoding threshold of low-density parity-check (LDPC) codes over the binary erasure channel (BEC) fulfills an upper bound depending only on the variable and check nodes with minimum distance 2. This bound is a consequence of…
In [1], K\"otter and Kschischang presented a new model for error correcting codes in network coding. The alphabet in this model is the subspace lattice of a given vector space, a code is a subset of this lattice and the used metric on this…