2,641 research outputs found
Asymptotically Good Additive Cyclic Codes Exist
Long quasi-cyclic codes of any fixed index have been shown to be
asymptotically good, depending on Artin primitive root conjecture in (A.
Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result
to construct good long additive cyclic codes on any extension of fixed degree
of the base field. Similarly self-dual double circulant codes, and self-dual
four circulant codes, have been shown to be good, also depending on Artin
primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and (
M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent
results, we can show that long cyclic codes are good over \F_q, for many
classes of 's. This is a partial solution to a fifty year old open problem
Low-complexity quantum codes designed via codeword-stabilized framework
We consider design of the quantum stabilizer codes via a two-step,
low-complexity approach based on the framework of codeword-stabilized (CWS)
codes. In this framework, each quantum CWS code can be specified by a graph and
a binary code. For codes that can be obtained from a given graph, we give
several upper bounds on the distance of a generic (additive or non-additive)
CWS code, and the lower Gilbert-Varshamov bound for the existence of additive
CWS codes. We also consider additive cyclic CWS codes and show that these codes
correspond to a previously unexplored class of single-generator cyclic
stabilizer codes. We present several families of simple stabilizer codes with
relatively good parameters.Comment: 12 pages, 3 figures, 1 tabl
Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs
A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling, and then regrouping the code coordinates. For any fixed (small) rate, and for a sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF(2)) as well. Although these concatenated codes lie below the Zyablov bound, they are still superior to previously known explicit constructions in the zero-rate neighborhood
Construction of isodual codes from polycirculant matrices
Double polycirculant codes are introduced here as a generalization of double
circulant codes. When the matrix of the polyshift is a companion matrix of a
trinomial, we show that such a code is isodual, hence formally self-dual.
Numerical examples show that the codes constructed have optimal or
quasi-optimal parameters amongst formally self-dual codes. Self-duality, the
trivial case of isoduality, can only occur over \F_2 in the double circulant
case. Building on an explicit infinite sequence of irreducible trinomials over
\F_2, we show that binary double polycirculant codes are asymptotically good
High-rate self-synchronizing codes
Self-synchronization under the presence of additive noise can be achieved by
allocating a certain number of bits of each codeword as markers for
synchronization. Difference systems of sets are combinatorial designs which
specify the positions of synchronization markers in codewords in such a way
that the resulting error-tolerant self-synchronizing codes may be realized as
cosets of linear codes. Ideally, difference systems of sets should sacrifice as
few bits as possible for a given code length, alphabet size, and
error-tolerance capability. However, it seems difficult to attain optimality
with respect to known bounds when the noise level is relatively low. In fact,
the majority of known optimal difference systems of sets are for exceptionally
noisy channels, requiring a substantial amount of bits for synchronization. To
address this problem, we present constructions for difference systems of sets
that allow for higher information rates while sacrificing optimality to only a
small extent. Our constructions utilize optimal difference systems of sets as
ingredients and, when applied carefully, generate asymptotically optimal ones
with higher information rates. We also give direct constructions for optimal
difference systems of sets with high information rates and error-tolerance that
generate binary and ternary self-synchronizing codes.Comment: 9 pages, no figure, 2 tables. Final accepted version for publication
in the IEEE Transactions on Information Theory. Material presented in part at
the International Symposium on Information Theory and its Applications,
Honolulu, HI USA, October 201
Asymmetric Quantum Codes: New Codes from Old
In this paper we extend to asymmetric quantum error-correcting codes (AQECC)
the construction methods, namely: puncturing, extending, expanding, direct sum
and the (u|u + v) construction. By applying these methods, several families of
asymmetric quantum codes can be constructed. Consequently, as an example of
application of quantum code expansion developed here, new families of
asymmetric quantum codes derived from generalized Reed-Muller (GRM) codes,
quadratic residue (QR), Bose-Chaudhuri-Hocquenghem (BCH), character codes and
affine-invariant codes are constructed.Comment: Accepted for publication Quantum Information Processin
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