Related papers: A Combination of Testability and Decodability by T…
In this paper we construct constant dimension space codes with prescribed minimum distance. There is an increased interest in space codes since a paper by Koetter and Kschischang were they gave an application in network coding. There is…
We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We…
We initiate a study of locally decodable codes with randomized encoding. Standard locally decodable codes are error correcting codes with a deterministic encoding function and a randomized decoding function, such that any desired message…
It was shown in \cite{GXY18} that the length $n$ of a $q$-ary linear locally recoverable code with distance $d\ge 5$ is upper bounded by $O(dq^3)$. Thus, it is a challenging problem to construct $q$-ary locally recoverable codes with…
In this paper, we prove a lower bound on the soundness of quantum locally testable codes under the distance balancing construction of Evra et al. arXiv:2004.07935 [quant-ph]. Our technical contribution is that the new soundness of the…
This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a…
Both completely positive and completely copositive maps stay decomposable under tensor powers, i.e under tensoring the linear map with itself. But are there other examples of maps with this property? We show that this is not the case: Any…
We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise…
Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic.…
We introduce a high-dimensional cubical complex, for any dimension t>0, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al…
It is now well known that the performance of a linear code $C$ under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for $C$. Several recent…
We introduce a simple two-player test which certifies that the players apply tensor products of Pauli $\sigma_X$ and $\sigma_Z$ observables on the tensor product of $n$ EPR pairs. The test has constant robustness: any strategy achieving…
A locally testable code (LTC) is an error-correcting code that has a property-tester. The tester reads $q$ bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter $q$ is…
Every ordered collection of sets in Euclidean space can be associated to a combinatorial code, which records the regions cut out by the sets in space. Given two ordered collections of sets, one can form a third collection in which the…
In this paper the ensemble of codes formed by a serial concatenation of a repetition code with multiple accumulators connected through random interleavers is considered. Based on finite length weight enumerators for these codes, asymptotic…
We construct 2-query, quasi-linear size probabilistically checkable proofs (PCPs) with arbitrarily small constant soundness, improving upon Dinur's 2-query quasi-linear size PCPs with soundness $1-\Omega(1)$. As an immediate corollary, we…
We analyze the list-decodability, and related notions, of random linear codes. This has been studied extensively before: there are many different parameter regimes and many different variants. Previous works have used complementary styles…
The problem of low complexity, close to optimal, channel decoding of linear codes with short to moderate block length is considered. It is shown that deep learning methods can be used to improve a standard belief propagation decoder,…
The error pattern correcting code (EPCC) can be constructed to provide a syndrome decoding table targeting the dominant error events of an inter-symbol interference channel at the output of the Viterbi detector. For the size of the syndrome…
Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity…