915,181 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
Cyclic LRC Codes and their Subfield Subcodes
We consider linear cyclic codes with the locality property, or locally
recoverable codes (LRC codes). A family of LRC codes that generalizes the
classical construction of Reed-Solomon codes was constructed in a recent paper
by I. Tamo and A. Barg (IEEE Transactions on Information Theory, no. 8, 2014;
arXiv:1311.3284). In this paper we focus on the optimal cyclic codes that arise
from the general construction. We give a characterization of these codes in
terms of their zeros, and observe that there are many equivalent ways of
constructing optimal cyclic LRC codes over a given field. We also study
subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish
several results about their locality and minimum distance.Comment: Submitted for publicatio
Lattice polytopes in coding theory
In this paper we discuss combinatorial questions about lattice polytopes
motivated by recent results on minimum distance estimation for toric codes. We
also prove a new inductive bound for the minimum distance of generalized toric
codes. As an application, we give new formulas for the minimum distance of
generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 figure
Some Applications of Coding Theory in Computational Complexity
Error-correcting codes and related combinatorial constructs play an important
role in several recent (and old) results in computational complexity theory. In
this paper we survey results on locally-testable and locally-decodable
error-correcting codes, and their applications to complexity theory and to
cryptography.
Locally decodable codes are error-correcting codes with sub-linear time
error-correcting algorithms. They are related to private information retrieval
(a type of cryptographic protocol), and they are used in average-case
complexity and to construct ``hard-core predicates'' for one-way permutations.
Locally testable codes are error-correcting codes with sub-linear time
error-detection algorithms, and they are the combinatorial core of
probabilistically checkable proofs
Cyclic LRC Codes, binary LRC codes, and upper bounds on the distance of cyclic codes
We consider linear cyclic codes with the locality property, or locally
recoverable codes (LRC codes). A family of LRC codes that generalize the
classical construction of Reed-Solomon codes was constructed in a recent paper
by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, no. 8, 2014). In this paper
we focus on optimal cyclic codes that arise from this construction. We give a
characterization of these codes in terms of their zeros, and observe that there
are many equivalent ways of constructing optimal cyclic LRC codes over a given
field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes)
and establish several results about their locality and minimum distance. The
locality parameter of a cyclic code is related to the dual distance of this
code, and we phrase our results in terms of upper bounds on the dual distance.Comment: 12pp., submitted for publication. An extended abstract of this
submission was posted earlier as arXiv:1502.01414 and was published in
Proceedings of the 2015 IEEE International Symposium on Information Theory,
Hong Kong, China, June 14-19, 2015, pp. 1262--126
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