494 research outputs found
Fast systematic encoding of multiplicity codes
We present quasi-linear time systematic encoding algorithms for multiplicity
codes. The algorithms have their origins in the fast multivariate interpolation
and evaluation algorithms of van der Hoeven and Schost (2013), which we
generalise to address certain Hermite-type interpolation and evaluation
problems. By providing fast encoding algorithms for multiplicity codes, we
remove an obstruction on the road to the practical application of the private
information retrieval protocol of Augot, Levy-dit-Vehel and Shikfa (2014)
Efficient Systematic Encoding of Non-binary VT Codes
Varshamov-Tenengolts (VT) codes are a class of codes which can correct a
single deletion or insertion with a linear-time decoder. This paper addresses
the problem of efficient encoding of non-binary VT codes, defined over an
alphabet of size . We propose a simple linear-time encoding method to
systematically map binary message sequences onto VT codewords. The method
provides a new lower bound on the size of -ary VT codes of length .Comment: This paper will appear in the proceedings of ISIT 201
Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes
In this paper, we establish a lemma in algebraic coding theory that
frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes,
algebraic geometry codes, and affine variety codes. Our lemma corresponds to
the non-systematic encoding of affine variety codes, and can be stated by
giving a canonical linear map as the composition of an extension through linear
feedback shift registers from a Grobner basis and a generalized inverse
discrete Fourier transform. We clarify that our lemma yields the error-value
estimation in the fast erasure-and-error decoding of a class of dual affine
variety codes. Moreover, we show that systematic encoding corresponds to a
special case of erasure-only decoding. The lemma enables us to reduce the
computational complexity of error-evaluation from O(n^3) using Gaussian
elimination to O(qn^2) with some mild conditions on n and q, where n is the
code length and q is the finite-field size.Comment: 37 pages, 1 column, 10 figures, 2 tables, resubmitted to IEEE
Transactions on Information Theory on Jan. 8, 201
A Low Complexity Algorithm and Architecture for Systematic Encoding of Hermitian Codes
We present an algorithm for systematic encoding of Hermitian codes. For a
Hermitian code defined over GF(q^2), the proposed algorithm achieves a run time
complexity of O(q^2) and is suitable for VLSI implementation. The encoder
architecture uses as main blocks q varying-rate Reed-Solomon encoders and
achieves a space complexity of O(q^2) in terms of finite field multipliers and
memory elements.Comment: 5 Pages, Accepted in IEEE International Symposium on Information
Theory ISIT 200
Information Sets of Multiplicity Codes
We here provide a method for systematic encoding of the Multiplicity codes
introduced by Kopparty, Saraf and Yekhanin in 2011. The construction is built
on an idea of Kop-party. We properly define information sets for these codes
and give detailed proofs of the validity of Kopparty's construction, that use
generating functions. We also give a complexity estimate of the associated
encoding algorithm.Comment: International Symposium on Information Theory, Jun 2015, Hong-Kong,
China. IEE
Fast Erasure-and-Error Decoding and Systematic Encoding of a Class of Affine Variety Codes
In this paper, a lemma in algebraic coding theory is established, which is
frequently appeared in the encoding and decoding for algebraic codes such as
Reed-Solomon codes and algebraic geometry codes. This lemma states that two
vector spaces, one corresponds to information symbols and the other is indexed
by the support of Grobner basis, are canonically isomorphic, and moreover, the
isomorphism is given by the extension through linear feedback shift registers
from Grobner basis and discrete Fourier transforms. Next, the lemma is applied
to fast unified system of encoding and decoding erasures and errors in a
certain class of affine variety codes.Comment: 6 pages, 2 columns, presented at The 34th Symposium on Information
Theory and Its Applications (SITA2011
Flexible and Low-Complexity Encoding and Decoding of Systematic Polar Codes
In this work, we present hardware and software implementations of flexible
polar systematic encoders and decoders. The proposed implementations operate on
polar codes of any length less than a maximum and of any rate. We describe the
low-complexity, highly parallel, and flexible systematic-encoding algorithm
that we use and prove its correctness. Our hardware implementation results show
that the overhead of adding code rate and length flexibility is little, and the
impact on operation latency minor compared to code-specific versions. Finally,
the flexible software encoder and decoder implementations are also shown to be
able to maintain high throughput and low latency.Comment: Submitted to IEEE Transactions on Communications, 201
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