9,829 research outputs found
Self-Repairing Codes for Distributed Storage - A Projective Geometric Construction
Self-Repairing Codes (SRC) are codes designed to suit the need of coding for
distributed networked storage: they not only allow stored data to be recovered
even in the presence of node failures, they also provide a repair mechanism
where as little as two live nodes can be contacted to regenerate the data of a
failed node. In this paper, we propose a new instance of self-repairing codes,
based on constructions of spreads coming from projective geometry. We study
some of their properties to demonstrate the suitability of these codes for
distributed networked storage.Comment: 5 pages, 2 figure
Quantum Synchronizable Codes From Finite Geometries
Quantum synchronizable error-correcting codes are special quantum error-correcting codes that are designed to correct both the effect of quantum noise on qubits and misalignment in block synchronization. It is known that, in principle, such a code can be constructed through a combination of a classical linear code and its subcode if the two are both cyclic and dual-containing. However, finding such classical codes that lead to promising quantum synchronizable error-correcting codes is not a trivial task. In fact, although there are two families of classical codes that are proved to produce quantum synchronizable codes with good minimum distances and highest possible tolerance against misalignment, their code lengths have been restricted to primes and Mersenne numbers. In this paper, examining the incidence vectors of projective spaces over the finite fields of characteristic 2, we give quantum synchronizable codes from cyclic codes whose lengths are not primes or Mersenne numbers. These projective geometric codes achieve good performance in quantum error correction and possess the best possible ability to recover synchronization, thereby enriching the variety of good quantum synchronizable codes. We also extend the current knowledge of cyclic codes in classical coding theory by explicitly giving generator polynomials of the finite geometric codes and completely characterizing the minimum weight nonzero codewords. In addition to the codes based on projective spaces, we carry out a similar analysis on the well-known cyclic codes from Euclidean spaces that are known to be majority logic decodable and determine their exact minimum distances
Small weight codewords of projective geometric codes II
The -ary linear code is defined as the row space of
the incidence matrix of -spaces and points of . It is
known that if is square, a codeword of weight exists that cannot be written as a linear combination
of at most rows of . Over the past few decades, researchers have
put a lot of effort towards proving that any codeword of smaller weight does
meet this property. We show that if is a composite prime
power, every codeword of up to weight is a linear combination of at most rows of
. We also generalise this result to the codes ,
which are defined as the -ary row span of the incidence matrix of -spaces
and -spaces, .Comment: 22 page
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
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