271 research outputs found
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
An upper bound of Singleton type for componentwise products of linear codes
We give an upper bound that relates the minimum weight of a nonzero
componentwise product of codewords from some given number of linear codes, with
the dimensions of these codes. Its shape is a direct generalization of the
classical Singleton bound.Comment: 9 pages; major improvements in v3: now works for an arbitrary number
of codes, and the low-weight codeword can be taken in product form; submitted
to IEEE Trans. Inform. Theor
Asymptotically good binary linear codes with asymptotically good self-intersection spans
If C is a binary linear code, let C^2 be the linear code spanned by
intersections of pairs of codewords of C. We construct an asymptotically good
family of binary linear codes such that, for C ranging in this family, the C^2
also form an asymptotically good family. For this we use algebraic-geometry
codes, concatenation, and a fair amount of bilinear algebra.
More precisely, the two main ingredients used in our construction are, first,
a description of the symmetric square of an odd degree extension field in terms
only of field operations of small degree, and second, a recent result of
Garcia-Stichtenoth-Bassa-Beelen on the number of points of curves on such an
odd degree extension field.Comment: 18 pages; v2->v3: expanded introduction and bibliography + various
minor change
Squares of matrix-product codes
The component-wise or Schur product of two linear error-correcting codes and over certain finite field is the linear code spanned by all component-wise products of a codeword in with a codeword in . When , we call the product the square of and denote it . Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrix-product codes, a general construction that allows to obtain new longer codes from several ``constituent'' codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known -construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain sense maximal with respect to matrix-product codes).This work is supported by the Danish Council for IndependentResearch: grant DFF-4002-00367, theSpanish Ministry of Economy/FEDER: grant RYC-2016-20208 (AEI/FSE/UE), the Spanish Ministry of Science/FEDER: grant PGC2018-096446-B-C21, and Junta de CyL (Spain): grant VA166G
Computation with finite fields
A technique for systematically generating representations of finite fields is presented. Relations which must be physically realized in order to implement a parallel arithmetic unit to add, multiply, and divide elements of finite fields of 2n elements are obtained. Finally, techniques for using a maximal length linear recurring sequence to modulate a radar transmitter and the means of extracting range information from the returned sequence are derived.*Operated with support from the U.S. Army, Navy and Air Force
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