82 research outputs found
Transitive and self-dual codes attaining the Tsfasman-Vladut-Zink bound
A major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good. In this correspondence-as a generalization of cyclic codes-the notion of transitive codes is introduced (see Definition 1.4 in Section I), and it is shown that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vladut-Zink bound over F-q, for all squares q = l(2). It is also shown that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower E-0 subset of E-1 subset of E-2 subset of center dot center dot center dot of function fields over F-q (with q = l(2)), where all extensions E-n/E-0 are Galois
Subquadratic time encodable codes beating the Gilbert-Varshamov bound
We construct explicit algebraic geometry codes built from the
Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for
alphabet sizes at least 192. Messages are identied with functions in certain
Riemann-Roch spaces associated with divisors supported on multiple places.
Encoding amounts to evaluating these functions at degree one places. By
exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we
devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and
1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list)
decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent.
If \omega = 2, as widely believed, the encoding and decoding runtimes are
respectively nearly linear and nearly quadratic. Prior to this work, encoding
(resp. decoding) time of code families beating the Gilbert-Varshamov bound were
quadratic (resp. cubic) or worse
Entanglement-assisted Quantum Codes from Algebraic Geometry Codes
Quantum error correcting codes play the role of suppressing noise and
decoherence in quantum systems by introducing redundancy. Some strategies can
be used to improve the parameters of these codes. For example, entanglement can
provide a way for quantum error correcting codes to achieve higher rates than
the one obtained via the traditional stabilizer formalism. Such codes are
called entanglement-assisted quantum (QUENTA) codes. In this paper, we use
algebraic geometry codes to construct several families of QUENTA codes via the
Euclidean and the Hermitian construction. Two of the families created have
maximal entanglement and have quantum Singleton defect equal to zero or one.
Comparing the other families with the codes with the respective quantum
Gilbert-Varshamov bound, we show that our codes have a rate that surpasses that
bound. At the end, asymptotically good towers of linear complementary dual
codes are used to obtain asymptotically good families of maximal entanglement
QUENTA codes. Furthermore, a simple comparison with the quantum
Gilbert-Varshamov bound demonstrates that using our construction it is possible
to create an asymptotically family of QUENTA codes that exceeds this bound.Comment: Some results in this paper were presented at the 2019 IEEE
International Symposium on Information Theor
Torsion Limits and Riemann-Roch Systems for Function Fields and Applications
The Ihara limit (or -constant) has been a central problem of study in
the asymptotic theory of global function fields (or equivalently, algebraic
curves over finite fields). It addresses global function fields with many
rational points and, so far, most applications of this theory do not require
additional properties. Motivated by recent applications, we require global
function fields with the additional property that their zero class divisor
groups contain at most a small number of -torsion points. We capture this by
the torsion limit, a new asymptotic quantity for global function fields. It
seems that it is even harder to determine values of this new quantity than the
Ihara constant. Nevertheless, some non-trivial lower- and upper bounds are
derived. Apart from this new asymptotic quantity and bounds on it, we also
introduce Riemann-Roch systems of equations. It turns out that this type of
equation system plays an important role in the study of several other problems
in areas such as coding theory, arithmetic secret sharing and multiplication
complexity of finite fields etc. Finally, we show how our new asymptotic
quantity, our bounds on it and Riemann-Roch systems can be used to improve
results in these areas.Comment: Accepted for publication in IEEE Transactions on Information Theory.
This is an extended version of our paper in Proceedings of 31st Annual IACR
CRYPTO, Santa Barbara, Ca., USA, 2011. The results in Sections 5 and 6 did
not appear in that paper. A first version of this paper has been widely
circulated since November 200
Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs
A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling, and then regrouping the code coordinates. For any fixed (small) rate, and for a sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF(2)) as well. Although these concatenated codes lie below the Zyablov bound, they are still superior to previously known explicit constructions in the zero-rate neighborhood
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