112 research outputs found
Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes
A lower bound on the number of uncorrectable errors of weight half the
minimum distance is derived for binary linear codes satisfying some condition.
The condition is satisfied by some primitive BCH codes, extended primitive BCH
codes, Reed-Muller codes, and random linear codes. The bound asymptotically
coincides with the corresponding upper bound for Reed-Muller codes and random
linear codes. By generalizing the idea of the lower bound, a lower bound on the
number of uncorrectable errors for weights larger than half the minimum
distance is also obtained, but the generalized lower bound is weak for large
weights. The monotone error structure and its related notion larger half and
trial set, which are introduced by Helleseth, Kl{\o}ve, and Levenshtein, are
mainly used to derive the bounds.Comment: 5 pages, to appear in ISIT 200
Universal fault-tolerant gates on concatenated stabilizer codes
It is an oft-cited fact that no quantum code can support a set of
fault-tolerant logical gates that is both universal and transversal. This no-go
theorem is generally responsible for the interest in alternative universality
constructions including magic state distillation. Widely overlooked, however,
is the possibility of non-transversal, yet still fault-tolerant, gates that
work directly on small quantum codes. Here we demonstrate precisely the
existence of such gates. In particular, we show how the limits of
non-transversality can be overcome by performing rounds of intermediate
error-correction to create logical gates on stabilizer codes that use no
ancillas other than those required for syndrome measurement. Moreover, the
logical gates we construct, the most prominent examples being Toffoli and
controlled-controlled-Z, often complete universal gate sets on their codes. We
detail such universal constructions for the smallest quantum codes, the 5-qubit
and 7-qubit codes, and then proceed to generalize the approach. One remarkable
result of this generalization is that any nondegenerate stabilizer code with a
complete set of fault-tolerant single-qubit Clifford gates has a universal set
of fault-tolerant gates. Another is the interaction of logical qubits across
different stabilizer codes, which, for instance, implies a broadly applicable
method of code switching.Comment: 18 pages + 5 pages appendix, 12 figure
Sparse Graph Codes for Quantum Error-Correction
We present sparse graph codes appropriate for use in quantum
error-correction. Quantum error-correcting codes based on sparse graphs are of
interest for three reasons. First, the best codes currently known for classical
channels are based on sparse graphs. Second, sparse graph codes keep the number
of quantum interactions associated with the quantum error correction process
small: a constant number per quantum bit, independent of the blocklength.
Third, sparse graph codes often offer great flexibility with respect to
blocklength and rate. We believe some of the codes we present are unsurpassed
by previously published quantum error-correcting codes.Comment: Version 7.3e: 42 pages. Extended version, Feb 2004. A shortened
version was resubmitted to IEEE Transactions on Information Theory Jan 20,
200
Mathematical structures for decoding projective geometry codes
Imperial Users onl
Variations of the McEliece Cryptosystem
Two variations of the McEliece cryptosystem are presented. The first one is
based on a relaxation of the column permutation in the classical McEliece
scrambling process. This is done in such a way that the Hamming weight of the
error, added in the encryption process, can be controlled so that efficient
decryption remains possible. The second variation is based on the use of
spatially coupled moderate-density parity-check codes as secret codes. These
codes are known for their excellent error-correction performance and allow for
a relatively low key size in the cryptosystem. For both variants the security
with respect to known attacks is discussed
Easily decoded error correcting codes
This thesis is concerned with the decoding aspect of linear block error-correcting codes. When, as in most practical situations, the decoder cost is limited an optimum code may be inferior in performance to a longer sub-optimum code' of the same rate. This consideration is a central theme of the thesis.
The best methods available for decoding short optimum codes and long B.C.H. codes are discussed, in some cases new decoding algorithms for the codes are introduced.
Hashim's "Nested" codes are then analysed. The method of nesting codes which was given by Hashim is shown to be optimum - but it is seen that the codes are less easily decoded than was previously thought.
"Conjoined" codes are introduced. It is shown how two codes with identical numbers of information bits may be "conjoined" to give a code with length and minimum distance equal to the sum of the respective parameters of the constituent codes but with the same number of information bits. A very simple decoding algorithm is given for the codes whereby each constituent codeword is decoded and then a decision is made as to the correct decoding. A technique is given for adding more codewords to conjoined codes without unduly increasing the decoder complexity.
Lastly, "Array" codes are described. They are formed by making parity checks over carefully chosen patterns of information bits arranged in a two-dimensional array. Various methods are given for choosing suitable patterns. Some of the resulting codes are self-orthogonal and certain of these have parameters close to the optimum for such codes. A method is given for adding more codewords to array codes, derived from a process of augmentation known for product codes
Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications
Coding; Communications; Engineering; Networks; Information Theory; Algorithm
Performance Analysis of Quantum Error-Correcting Codes via MacWilliams Identities
One of the main challenges for an efficient implementation of quantum
information technologies is how to counteract quantum noise. Quantum error
correcting codes are therefore of primary interest for the evolution towards
quantum computing and quantum Internet. We analyze the performance of
stabilizer codes, one of the most important classes for practical
implementations, on both symmetric and asymmetric quantum channels. To this
aim, we first derive the weight enumerator (WE) for the undetectable errors of
stabilizer codes based on the quantum MacWilliams identities. The WE is then
used to evaluate the error rate of quantum codes under maximum likelihood
decoding or, in the case of surface codes, under minimum weight perfect
matching (MWPM) decoding. Our findings lead to analytical formulas for the
performance of generic stabilizer codes, including the Shor code, the Steane
code, as well as surface codes. For example, on a depolarizing channel with
physical error rate it is found that the logical error rate
is asymptotically for the
Shor code, for the
Steane code, for the surface
code, and for the surface
code.Comment: 25 pages, 5 figures, submitted to an IEEE journal. arXiv admin note:
substantial text overlap with arXiv:2302.1301
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