1,970 research outputs found
Maximum Weight Spectrum Codes
In the recent work \cite{shi18}, a combinatorial problem concerning linear
codes over a finite field \F_q was introduced. In that work the authors
studied the weight set of an linear code, that is the set of non-zero
distinct Hamming weights, showing that its cardinality is upper bounded by
. They showed that this bound was sharp in the case ,
and in the case . They conjectured that the bound is sharp for every
prime power and every positive integer . In this work quickly
establish the truth of this conjecture. We provide two proofs, each employing
different construction techniques. The first relies on the geometric view of
linear codes as systems of projective points. The second approach is purely
algebraic. We establish some lower bounds on the length of codes that satisfy
the conjecture, and the length of the new codes constructed here are discussed.Comment: 19 page
A proof of the Etzion-Silberstein conjecture for monotone and MDS-constructible Ferrers diagrams
Ferrers diagram rank-metric codes were introduced by Etzion and Silberstein
in 2009. In their work, they proposed a conjecture on the largest dimension of
a space of matrices over a finite field whose nonzero elements are supported on
a given Ferrers diagram and all have rank lower bounded by a fixed positive
integer . Since stated, the Etzion-Silberstein conjecture has been verified
in a number of cases, often requiring additional constraints on the field size
or on the minimum rank in dependence of the corresponding Ferrers diagram.
As of today, this conjecture still remains widely open. Using modular methods,
we give a constructive proof of the Etzion-Silberstein conjecture for the class
of strictly monotone Ferrers diagrams, which does not depend on the minimum
rank and holds over every finite field. In addition, we leverage on the
last result to also prove the conjecture for the class of MDS-constructible
Ferrers diagrams, without requiring any restriction on the field size.Comment: 21 page
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