We consider the problem of secure distributed ma-trix multiplication in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. We show how to construct polynomial schemes for the outer product partitioning which take advantage of the user's ability to precompute, and provide bounds for our technique. We show that precomputation allows for a reduction in the order of the time complexity for the cases where the number of colluding servers is a fixed percentage of the number of servers. Furthermore, with precomputation, any percentage (less than 100%) of collusions can be tolerated, compared to the upper limit of 50% for the case without precomputation.

A constant-dimension code (CDC) is a set of subspaces of constant dimension in a common vector space with upper bounded pairwise intersection. We improve and generalize two constructions for CDCs, theimproved linkage constructionand theparallel linkage construction, to thegeneralized linkage constructionand themultiblock generalized linkage constructionwhich in turn yield many improved lower bounds for the cardinalities of CDCs; a quantity not known in general.

We consider the problem of secure distributed matrix multiplication (SDMM) in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. We construct polynomial codes for SDMM by studying a recently introduced combinatorial tool called the degree table. Maximizing the download rate of a polynomial code for SDMM is equivalent to minimizing N, the number of distinct elements in the corresponding degree table. We propose new constructions of degree tables with a low number of distinct elements. These new constructions lead to a general family of polynomial codes for SDMM, which we call GASP,. (Gap Additive Secure Polynomial codes) parametrized by an integer r. GASProutperforms all previously known polynomial codes for SDMM. We also present lower bounds on N and show that GASPrachieves the lower bounds in the case of no server collusion.

We prove upper bounds for the cardinality of constant dimension codes (CDC) which contain a lifted maximum rank distance (LMRD) code as a subset. Thereby we cover all parameters fulfilling k <; 3d/2, where k is the codeword dimension and d is the minimum subspace distance. The proofs of the bounds additionally show that an LMRD code L can be unioned with a CDC C (of fitting parameters) without violating the subspace distance condition iff each codeword of C intersects the special subspace of L in at least dimension d/2. This connection is used to find the new largest and sometimes bound achieving CDCs for small parameters.

One of the main problems of the research area of network coding is to compute good lower and upper bounds of the achievable cardinality of so-called subspace codes in Pq(n), i.e., the set of subspaces of Fqn, for a given minimal distance. Here we generalize a construction of Etzion and Silberstein to a wide range of parameters. This construction, named coset construction, improves or attains several of the previously best-known subspace code sizes and attains the maximum-rank distance bound for an infinite family of parameters.