40 research outputs found
The equivalence of linear codes implies semi-linear equivalence
The version of record is available online at: 10.1007/s10623-022-01055-xWe prove that if two linear codes are equivalent then they are semi-linearly equivalent. We also prove that if two additive MDS codes over a field are equivalent then they are additively equivalent.Peer ReviewedPostprint (author's final draft
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
New scattered subspaces in higher dimensions
Over the past few decades, there has been extensive research on scattered
subspaces, partly because of their link to MRD codes. These subspaces can be
characterized using linearized polynomials over finite fields. Within this
context, scattered sequences extend the concept of scattered polynomials and
can be viewed as geometric equivalents of exceptional MRD codes. Up to now,
only scattered sequences of orders one and two have been developed. However,
this paper presents an infinite series of exceptional scattered sequences of
any order beyond two which correspond to scattered subspaces that cannot be
obtained as direct sum of scattered subspaces in smaller dimensions. The paper
also addresses equivalence concerns within this framework.Comment: arXiv admin note: substantial text overlap with arXiv:2306.0344