2,351 research outputs found
Quasi-cyclic subcodes of cyclic codes
We completely characterize possible indices of quasi-cyclic subcodes in a
cyclic code for a very broad class of cyclic codes. We present enumeration
results for quasi-cyclic subcodes of a fixed index and show that the problem of
enumeration is equivalent to enumeration of certain vector subspaces in finite
fields. In particular, we present enumeration results for quasi-cyclic subcodes
of the simplex code and duals of certain BCH codes. Our results are based on
the trace representation of cyclic codes
Magnitude cohomology
Magnitude homology was introduced by Hepworth and Willerton in the case of
graphs, and was later extended by Leinster and Shulman to metric spaces and
enriched categories. Here we introduce the dual theory, magnitude cohomology,
which we equip with the structure of an associative unital graded ring. Our
first main result is a 'recovery theorem' showing that the magnitude cohomology
ring of a finite metric space completely determines the space itself. The
magnitude cohomology ring is non-commutative in general, for example when
applied to finite metric spaces, but in some settings it is commutative, for
example when applied to ordinary categories. Our second main result explains
this situation by proving that the magnitude cohomology ring of an enriched
category is graded-commutative whenever the enriching category is cartesian. We
end the paper by giving complete computations of magnitude cohomology rings for
several large classes of graphs.Comment: 27 page
Optimal Binary Locally Repairable Codes via Anticodes
This paper presents a construction for several families of optimal binary
locally repairable codes (LRCs) with small locality (2 and 3). This
construction is based on various anticodes. It provides binary LRCs which
attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal
with respect to the Griesmer bound
The Weights in MDS Codes
The weights in MDS codes of length n and dimension k over the finite field
GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with
parameters given by the MDS conjecture are shown to contain all k weights in
the range n-k+1 to n. The proof uses the covering radius of the dual codeComment: 5 pages, submitted to IEEE Trans. IT. This version 2 is the revised
version after the refereeing process. Accepted for publicatio
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