49,519 research outputs found
On codes satisfying the double chain condition
AbstractThe double chain condition is described. A number of bounds on the length and weight hierarchy of codes satisfying the double chain condition are given. Constructions of codes satisfying the double chain condition and with trellis complexity 1 or 2 are given
X-code: MDS array codes with optimal encoding
We present a new class of MDS (maximum distance separable) array codes of size nĂn (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns
Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions
Polar transforms are central operations in the study of polar codes. This
paper examines polar transforms for non-stationary memoryless sources on
possibly infinite source alphabets. This is the first attempt of source
polarization analysis over infinite alphabets. The source alphabet is defined
to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar
transform based on the group. Defining erasure distributions based on the
normal subgroup structure, we give recursive formulas of the polar transform
for our proposed erasure distributions. As a result, the recursive formulas
lead to concrete examples of multilevel source polarization with countably
infinite levels when the group is locally cyclic. We derive this result via
elementary techniques in lattice theory.Comment: 12 pages, 1 figure, a short version has been accepted by the 2019
IEEE International Symposium on Information Theory (ISIT2019
An application of Khovanov homology to quantum codes
We use Khovanov homology to define families of LDPC quantum error-correcting
codes: unknot codes with asymptotical parameters
[[3^(2l+1)/sqrt(8{\pi}l);1;2^l]]; unlink codes with asymptotical parameters
[[sqrt(2/2{\pi}l)6^l;2^l;2^l]] and (2,l)-torus link codes with asymptotical
parameters [[n;1;d_n]] where d_n>\sqrt(n)/1.62.Comment: 20 page
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