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 the inner product partition for matrices A and B. Our central technique relies on encoding matrices A and B in a Hermitian code and its dual code, respectively. We present the Hermitian Algebraic (HerA) scheme, which employs Hermitian codes and characterizes the partitioning and security capacities given entries of matrices belonging to a finite field with q2elements. We showcase that this scheme performs the secure distributed matrix multiplication in a significantly smaller finite field and expands security allowances compared to the existing results in the literature.

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.

Consider a directed graph (digraph) in which vertices are assigned color sets, and two vertices are connected if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head. We seek to determine the smallest possible size of the union of the color sets that allows for such a digraph representation. To address this problem, we introduce the new notion of a directed intersection representation of a digraph, and show that it is well-defined for all directed acyclic graphs (DAGs). We then proceed to introduce the directed intersection number (DIN), the smallest number of colors needed to represent a DAG. Our main results are upper bounds on the DIN of DAGs based on what we call the longest terminal path decomposition of the vertex set, and constructive lower bounds.

Consider a directed graph (digraph) in which two user vertices are connected if and only if they share at least one unit of common information content and the head vertex has a strictly smaller content than the tail. We seek to estimate the smallest possible global information content that can explain the observed digraph topology. To address this problem, we introduce the new notion of a directed intersection representation of a digraph, and show that it is well-defined for all directed acyclic graphs (DAGs). We then proceed to describe the directed intersection number (DIN), the smallest number of information units needed to represent the DAG. Our main result is a nontrivial upper bound on the DIN number of DAGs based on the longest terminal path decomposition of the vertex set. In addition, we compute the exact values of the DIN number for several simple yet relevant families of connected DAGs and construct digraphs that have near-optimal DIN values.

In this work we explore a family of metrics over a finite field Fqwhich respect the support of vectors. We show how these metrics can be obtained from the edge-weighted Hamming cube and, based on this representation we give a description of the group of linear isometries (for q > 2). Next we introduce the concept of conditional sum of metrics and determine what are the conditions that, out of two metrics respecting the support, gives rise to a new such metric. Finally we introduce the labeled-poset block metrics, a new family of metrics which respects support of vectors, filling a gap existing in the known universe of such metrics. For this family we give a full description of the group of linear isometries and determine sufficient conditions for the existence of a MacWilliams' identity.

Given a finite directed graph with n vertices, we define a metric dG on Fnq, where Fq is the finite field with q elements. The weight of a word is defined as the number of vertices that can be reached by a directed path from a vertex within the support of the vector. Two canonical forms, which do not affect the metric, are given to each graph. Based on these forms we characterize each such metric. We further use these forms to prove that two graphs with different canonical forms yield different metrics. Efficient algorithms to check if a set of metric weights define a metric based on a graph are given. We provide tight bounds on the number of metric weights required to reconstruct the metric. Furthermore, we give a complete description of the group of linear isometries of the graph metrics and a characterization of the graphs for which every linear code admits a G-canonical decomposition. Considering those graphs, we are able to derive an expression of the packing radius of linear codes in such metric spaces. Finally, given a directed graph which determines a hierarchical poset, we present sufficient and necessary conditions to ensure the validity of the MacWilliams identity and the MacWilliams extension property.

In this paper, we consider metrics determined by hierarchical posets and give explicit formulas for the main parameters of a linear code: the minimum distance and the packing, covering, and Chebyshev radii of a code. We also present ten characterizations of hierarchical poset metrics, including new characterizations and simple new proofs to the known ones.

Considering metrics based on finite directed graph, introduced by Etzion and Firer, we characterize the graphs such that every linear code admits a G-canonical decomposition. This decomposition will play an important role in this work, since it will be the main tool to give a sufficient condition for a finite directed graph to satisfy both the MacWilliams Identity and the MacWilliams Extension Property.