On products and powers of linear codes under componentwise multiplication

In this text we develop the formalism of products and powers of linear codes under componentwise multiplication. As an expanded version of the author's talk at AGCT-14, focus is put mostly on basic properties and descriptive statements that could otherwise probably not fit in a regular research paper. On the other hand, more advanced results and applications are only quickly mentioned with references to the literature. We also point out a few open problems. Our presentation alternates between two points of view, which the theory intertwines in an essential way: that of combinatorial coding, and that of algebraic geometry. In appendices that can be read independently, we investigate topics in multilinear algebra over finite fields, notably we establish a criterion for a symmetric multilinear map to admit a symmetric algorithm, or equivalently, for a symmetric tensor to decompose as a sum of elementary symmetric tensors.Comment: 75 pages; expanded version of a talk at AGCT-14 (Luminy), to appear in vol. 637 of Contemporary Math., AMS, Apr. 2015; v3: minor typos corrected in the final "open questions" sectio

X-code: MDS array codes with optimal encoding

We present a new class of MDS (maximum distance separable) array codes of size n×n (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns

Linear sets in the projective line over the endomorphism ring of a finite field

Let $\mathrm{PG}(1,E)$ be the projective line over the endomorphism ring $E=End_q({\mathbb F}_{q^t})$ of the $\mathbb F_q$-vector space ${\mathbb F}_{q^t}$. As is well known there is a bijection $\Psi:\mathrm{PG}(1,E)\rightarrow{\cal G}_{2t,t,q}$ with the Grassmannian of the $(t-1)$-subspaces in $\mathrm{PG}(2t-1,q)$. In this paper along with any $\mathbb F_q$-linear set $L$ of rank $t$ in $\mathrm{PG}(1,q^t)$, determined by a $(t-1)$-dimensional subspace $T^\Psi$ of $\mathrm{PG}(2t-1,q)$, a subset $L_T$ of $\mathrm{PG}(1,E)$ is investigated. Some properties of linear sets are expressed in terms of the projective line over the ring $E$. In particular the attention is focused on the relationship between $L_T$ and the set $L'_T$, corresponding via $\Psi$ to a collection of pairwise skew $(t-1)$-dimensional subspaces, with $T\in L'_T$, each of which determine $L$. This leads among other things to a characterization of the linear sets of pseudoregulus type. It is proved that a scattered linear set $L$ related to $T\in\mathrm{PG}(1,E)$ is of pseudoregulus type if and only if there exists a projectivity $\varphi$ of $\mathrm{PG}(1,E)$ such that $L_T^\varphi=L'_T$