218 research outputs found
Higher Hamming weights for locally recoverable codes on algebraic curves
We study the locally recoverable codes on algebraic curves. In the first part
of this article, we provide a bound of generalized Hamming weight of these
codes. Whereas in the second part, we propose a new family of algebraic
geometric LRC codes, that are LRC codes from Norm-Trace curve. Finally, using
some properties of Hermitian codes, we improve the bounds of distance proposed
in [1] for some Hermitian LRC codes.
[1] A. Barg, I. Tamo, and S. Vlladut. Locally recoverable codes on algebraic
curves. arXiv preprint arXiv:1501.04904, 2015
List Decoding Tensor Products and Interleaved Codes
We design the first efficient algorithms and prove new combinatorial bounds
for list decoding tensor products of codes and interleaved codes. We show that
for {\em every} code, the ratio of its list decoding radius to its minimum
distance stays unchanged under the tensor product operation (rather than
squaring, as one might expect). This gives the first efficient list decoders
and new combinatorial bounds for some natural codes including multivariate
polynomials where the degree in each variable is bounded. We show that for {\em
every} code, its list decoding radius remains unchanged under -wise
interleaving for an integer . This generalizes a recent result of Dinur et
al \cite{DGKS}, who proved such a result for interleaved Hadamard codes
(equivalently, linear transformations). Using the notion of generalized Hamming
weights, we give better list size bounds for {\em both} tensoring and
interleaving of binary linear codes. By analyzing the weight distribution of
these codes, we reduce the task of bounding the list size to bounding the
number of close-by low-rank codewords. For decoding linear transformations,
using rank-reduction together with other ideas, we obtain list size bounds that
are tight over small fields.Comment: 32 page
Product Construction of Affine Codes
Binary matrix codes with restricted row and column weights are a desirable
method of coded modulation for power line communication. In this work, we
construct such matrix codes that are obtained as products of affine codes -
cosets of binary linear codes. Additionally, the constructions have the
property that they are systematic. Subsequently, we generalize our construction
to irregular product of affine codes, where the component codes are affine
codes of different rates.Comment: 13 pages, to appear in SIAM Journal on Discrete Mathematic
Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth's second tower
Asymptotically good sequences of ramp secret sharing schemes were given in
[Asymptotically good ramp secret sharing schemes, arXiv:1502.05507] by using
one-point algebraic geometric codes defined from asymptotically good towers of
function fields. Their security is given by the relative generalized Hamming
weights of the corresponding codes. In this paper we demonstrate how to obtain
refined information on the RGHWs when the codimension of the codes is small.
For general codimension, we give an improved estimate for the highest RGHW
Strong Singleton type upper bounds for linear insertion-deletion codes
The insertion-deletion codes was motivated to correct the synchronization
errors. In this paper we prove several Singleton type upper bounds on the
insdel distances of linear insertion-deletion codes, based on the generalized
Hamming weights and the formation of minimum Hamming weight codewords. Our
bound are stronger than some previous known bounds. These upper bounds are
valid for any fixed ordering of coordinate positions. We apply these upper
bounds to some binary cyclic codes and binary Reed-Muller codes with any
coordinate ordering, and some binary Reed-Muller codes and one
algebraic-geometric code with certain special coordinate ordering.Comment: 22 pages, references update
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