723 research outputs found
Improved Nearly-MDS Expander Codes
A construction of expander codes is presented with the following three
properties:
(i) the codes lie close to the Singleton bound, (ii) they can be encoded in
time complexity that is linear in their code length, and (iii) they have a
linear-time bounded-distance decoder.
By using a version of the decoder that corrects also erasures, the codes can
replace MDS outer codes in concatenated constructions, thus resulting in
linear-time encodable and decodable codes that approach the Zyablov bound or
the capacity of memoryless channels. The presented construction improves on an
earlier result by Guruswami and Indyk in that any rate and relative minimum
distance that lies below the Singleton bound is attainable for a significantly
smaller alphabet size.Comment: Part of this work was presented at the 2004 IEEE Int'l Symposium on
Information Theory (ISIT'2004), Chicago, Illinois (June 2004). This work was
submitted to IEEE Transactions on Information Theory on January 21, 2005. To
appear in IEEE Transactions on Information Theory, August 2006. 12 page
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
Linear-time list recovery of high-rate expander codes
We show that expander codes, when properly instantiated, are high-rate list
recoverable codes with linear-time list recovery algorithms. List recoverable
codes have been useful recently in constructing efficiently list-decodable
codes, as well as explicit constructions of matrices for compressive sensing
and group testing. Previous list recoverable codes with linear-time decoding
algorithms have all had rate at most 1/2; in contrast, our codes can have rate
for any . We can plug our high-rate codes into a
construction of Meir (2014) to obtain linear-time list recoverable codes of
arbitrary rates, which approach the optimal trade-off between the number of
non-trivial lists provided and the rate of the code. While list-recovery is
interesting on its own, our primary motivation is applications to
list-decoding. A slight strengthening of our result would implies linear-time
and optimally list-decodable codes for all rates, and our work is a step in the
direction of solving this important problem
The Small Stellated Dodecahedron Code and Friends
We explore a distance-3 homological CSS quantum code, namely the small
stellated dodecahedron code, for dense storage of quantum information and we
compare its performance with the distance-3 surface code. The data and ancilla
qubits of the small stellated dodecahedron code can be located on the edges
resp. vertices of a small stellated dodecahedron, making this code suitable for
3D connectivity. This code encodes 8 logical qubits into 30 physical qubits
(plus 22 ancilla qubits for parity check measurements) as compared to 1 logical
qubit into 9 physical qubits (plus 8 ancilla qubits) for the surface code. We
develop fault-tolerant parity check circuits and a decoder for this code,
allowing us to numerically assess the circuit-based pseudo-threshold.Comment: 19 pages, 14 figures, comments welcome! v2 includes updates which
conforms with the journal versio
Codes for protection from synchronization loss and additive errors
Codes for protection from synchronization loss and additive error
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