68,385 research outputs found
Concatenated Quantum Codes Constructible in Polynomial Time: Efficient Decoding and Error Correction
A method for concatenating quantum error-correcting codes is presented. The
method is applicable to a wide class of quantum error-correcting codes known as
Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate
in the Shannon theoretic sense and that are decodable in polynomial time are
presented. The rate is the highest among those known to be achievable by CSS
codes. Moreover, the best known lower bound on the greatest minimum distance of
codes constructible in polynomial time is improved for a wide range.Comment: 16 pages, 3 figures. Ver.4: Title changed. Ver.3: Due to a request of
the AE of the journal, the present version has become a combination of
(thoroughly revised) quant-ph/0610194 and the former quant-ph/0610195.
Problem formulations of polynomial complexity are strictly followed. An
erroneous instance of a lower bound on minimum distance was remove
Polar Codes with exponentially small error at finite block length
We show that the entire class of polar codes (up to a natural necessary
condition) converge to capacity at block lengths polynomial in the gap to
capacity, while simultaneously achieving failure probabilities that are
exponentially small in the block length (i.e., decoding fails with probability
for codes of length ). Previously this combination
was known only for one specific family within the class of polar codes, whereas
we establish this whenever the polar code exhibits a condition necessary for
any polarization.
Our results adapt and strengthen a local analysis of polar codes due to the
authors with Nakkiran and Rudra [Proc. STOC 2018]. Their analysis related the
time-local behavior of a martingale to its global convergence, and this allowed
them to prove that the broad class of polar codes converge to capacity at
polynomial block lengths. Their analysis easily adapts to show exponentially
small failure probabilities, provided the associated martingale, the ``Arikan
martingale'', exhibits a corresponding strong local effect. The main
contribution of this work is a much stronger local analysis of the Arikan
martingale. This leads to the general result claimed above.
In addition to our general result, we also show, for the first time, polar
codes that achieve failure probability for any
while converging to capacity at block length polynomial in the gap to capacity.
Finally we also show that the ``local'' approach can be combined with any
analysis of failure probability of an arbitrary polar code to get essentially
the same failure probability while achieving block length polynomial in the gap
to capacity.Comment: 17 pages, Appeared in RANDOM'1
On deep holes of generalized Reed-Solomon codes
Determining deep holes is an important topic in decoding Reed-Solomon codes.
In a previous paper [8], we showed that the received word is a deep hole of
the standard Reed-Solomon codes if its Lagrange interpolation
polynomial is the sum of monomial of degree and a polynomial of degree at
most . In this paper, we extend this result by giving a new class of deep
holes of the generalized Reed-Solomon codes.Comment: 5 page
Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes
Cyclic, negacyclic and constacyclic codes are part of a larger class of codes
called polycyclic codes; namely, those codes which can be viewed as ideals of a
factor ring of a polynomial ring. The structure of the ambient ring of
polycyclic codes over GR(p^a,m) and generating sets for its ideals are
considered. Along with some structure details of the ambient ring, the
existance of a certain type of generating set for an ideal is proven.Comment: arXiv admin note: text overlap with arXiv:0906.400
On the Construction of Prefix-Free and Fix-Free Codes with Specified Codeword Compositions
We investigate the construction of prefix-free and fix-free codes with
specified codeword compositions. We present a polynomial time algorithm which
constructs a fix-free code with the same codeword compositions as a given code
for a special class of codes called distinct codes. We consider the
construction of optimal fix-free codes which minimizes the average codeword
cost for general letter costs with uniform distribution of the codewords and
present an approximation algorithm to find a near optimal fix-free code with a
given constant cost
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