We introduce a formal framework to study the multiple unicast problem for a coded network in which the network code is linear over a finite field and fixed. We show that the problem corresponds to an interference alignment problem over a finite field. In this context, we establish an outer bound for the achievable rate region and provide examples of networks where the bound is sharp. We finally give evidence of the crucial role played by the field characteristic in the problem.

We consider the problem of secure distributed matrix multiplication (SDMM), where a user has two matrices and wishes to compute their product with the help of N honest but curious servers under the security constraint that any information about either A or B is not leaked to any server. This paper presents a new scheme that considers a grid product partition for matrices A and B, which achieves an upload cost significantly lower than the existing results in the literature. Also, it significantly reduces the recovery threshold compared to the PolyDot codes presented for grid partition when T > 0. Since the grid partition is a general partition that incorporates the inner and outer ones, it turns out that the communication load of the proposed scheme matches the best-known protocols for those extreme cases.

A broad range of scientific simulations involve solving large-scale computationally expensive linear systems of equations. Iterative solvers are typically preferred over direct methods when it comes to large systems due to their lower memory requirements and shorter execution times. However, selecting the appropriate iterative solver is problem-specific and dependent on the type and symmetry of the coefficient matrix. Gauss-Seidel (GS) is an iterative method for solving linear systems that are either strictly diagonally dominant or symmetric positive definite. This technique is an improved version of Jacobi and typically converges in fewer iterations. However, the sequential nature of this algorithm complicates the parallel extraction. In fact, most parallel derivatives of GS rely on the sparsity pattern of the coefficient matrix and require matrix reordering or domain decomposition. In this article, we introduce a new algorithm that exploits the convergence property of GS and adapts the parallel structure of Jacobi. The proposed method works for both dense and sparse systems and is straightforward to implement. We have examined the performance of our method on multicore and many-core architectures. Experimental results demonstrate the superior performance of the proposed algorithm compared with GS and Jacobi. Additionally, performance comparison with built-in Krylov solvers in MATLAB showed that in terms of time per iteration, Krylov methods perform faster on CPUs, but our approach is significantly better when executed on GPUs. Lastly, we apply our method to solve the power flow problem, and the results indicate a significant improvement in runtime, reaching up to 87 times faster speed compared with GS.

We would like to thank Mahdieh Hakimi Poroch and Ali Asghar Talebi for pointing out two errors inProposition 28andTheorem 29of the original paper[1]. The correct formulation for these two statements is as follows.

A constant dimension code consists of a set of k-dimensional subspaces of \BBF qn. Orbit codes are constant dimension codes which are defined as orbits of a subgroup of the general linear group, acting on the set of all subspaces of \BBF qn. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper, we show how orbit codes can be seen as an analog of linear codes in the block coding case. We investigate how the structure of cyclic orbit codes can be utilized to compute the minimum distance and cardinality of a given code and propose different decoding procedures for a particular subclass of cyclic orbit codes.

Orbit codes are a family of codes applicable for communications on a random linear network coding channel. The paper focuses on the classification of these codes. We start by classifying the conjugacy classes of cyclic subgroups of the general linear group. As a result, we are able to focus the study of cyclic orbit codes to a restricted family of them.

We introduce a new class of constant dimension codes called orbit codes. The basic properties of these codes are derived. It will be shown that many of the known families of constant dimension codes in the literature are actually orbit codes.

In this paper we introduce the class of spread codes for the use in random network coding. Spread codes are based on the construction of spreads in finite projective geometry. The major contribution of the paper is an efficient decoding algorithm of spread codes up to half the minimum distance.