5,326 research outputs found
A note on the minimum distance of quantum LDPC codes
We provide a new lower bound on the minimum distance of a family of quantum
LDPC codes based on Cayley graphs proposed by MacKay, Mitchison and
Shokrollahi. Our bound is exponential, improving on the quadratic bound of
Couvreur, Delfosse and Z\'emor. This result is obtained by examining a family
of subsets of the hypercube which locally satisfy some parity conditions
Small polygons and toric codes
We describe two different approaches to making systematic classifications of
plane lattice polygons, and recover the toric codes they generate, over small
fields, where these match or exceed the best known minimum distance. This
includes a [36,19,12]-code over F_7 whose minimum distance 12 exceeds that of
all previously known codes.Comment: 9 pages, 4 tables, 3 figure
Construction of Rational Surfaces Yielding Good Codes
In the present article, we consider Algebraic Geometry codes on some rational
surfaces. The estimate of the minimum distance is translated into a point
counting problem on plane curves. This problem is solved by applying the upper
bound "\`a la Weil" of Aubry and Perret together with the bound of Homma and
Kim for plane curves. The parameters of several codes from rational surfaces
are computed. Among them, the codes defined by the evaluation of forms of
degree 3 on an elliptic quadric are studied. As far as we know, such codes have
never been treated before. Two other rational surfaces are studied and very
good codes are found on them. In particular, a [57,12,34] code over
and a [91,18,53] code over are discovered, these
codes beat the best known codes up to now.Comment: 20 pages, 7 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,
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