77,864 research outputs found
Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes
We discuss error-correction properties for families of quantum low-density
parity check (LDPC) codes with relative distance that tends to zero in the
limit of large blocklength. In particular, we show that any family of LDPC
codes, quantum or classical, where distance scales as a positive power of the
block length, , , can correct all errors with
certainty if the error rate per (qu)bit is sufficiently small. We specifically
analyze the case of LDPC version of the quantum hypergraph-product codes
recently suggested by Tillich and Z\'emor. These codes are a finite-rate
generalization of the toric codes, and, for sufficiently large quantum
computers, offer an advantage over the toric codes.Comment: 4.5 pages, 1 figur
Spin glass reflection of the decoding transition for quantum error correcting codes
We study the decoding transition for quantum error correcting codes with the
help of a mapping to random-bond Wegner spin models.
Families of quantum low density parity-check (LDPC) codes with a finite
decoding threshold lead to both known models (e.g., random bond Ising and
random plaquette gauge models) as well as unexplored earlier generally
non-local disordered spin models with non-trivial phase diagrams. The decoding
transition corresponds to a transition from the ordered phase by proliferation
of extended defects which generalize the notion of domain walls to non-local
spin models. In recently discovered quantum LDPC code families with finite
rates the number of distinct classes of such extended defects is exponentially
large, corresponding to extensive ground state entropy of these codes.
Here, the transition can be driven by the entropy of the extended defects, a
mechanism distinct from that in the local spin models where the number of
defect types (domain walls) is always finite.Comment: 15 pages, 2 figure
Elias Bound for General Distances and Stable Sets in Edge-Weighted Graphs
This paper presents an extension of the Elias bound on the minimum distance
of codes for discrete alphabets with general, possibly infinite-valued,
distances. The bound is obtained by combining a previous extension of the Elias
bound, introduced by Blahut, with an extension of a bound previously introduced
by the author which builds upon ideas of Gallager, Lov\'asz and Marton. The
result can in fact be interpreted as a unification of the Elias bound and of
Lov\'asz's bound on graph (or zero-error) capacity, both being recovered as
particular cases of the one presented here. Previous extensions of the Elias
bound by Berlekamp, Blahut and Piret are shown to be included as particular
cases of our bound. Applications to the reliability function are then
discussed.Comment: Accepted, IEEE Transaction on Information Theor
On spectra of BCH codes
Derives an estimate for the error term in the binomial approximation of spectra of BCH codes. This estimate asymptotically improves on the bounds by Sidelnikov (1971), Kasami et al. (1985), and Sole (1990)
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