12,678 research outputs found
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
New Quantum Codes from Evaluation and Matrix-Product Codes
Stabilizer codes obtained via CSS code construction and Steane's enlargement
of subfield-subcodes and matrix-product codes coming from generalized
Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes
with good quantum parameters are supplied, in particular, some binary codes of
lengths 127 and 128 improve the parameters of the codes in
http://www.codetables.de. Moreover, non-binary codes are presented either with
parameters better than or equal to the quantum codes obtained from BCH codes by
La Guardia or with lengths that can not be reached by them
Explicit Subcodes of Reed-Solomon Codes that Efficiently Achieve List Decoding Capacity
In this paper, we introduce a novel explicit family of subcodes of
Reed-Solomon (RS) codes that efficiently achieve list decoding capacity with a
constant output list size. Our approach builds upon the idea of large linear
subcodes of RS codes evaluated on a subfield, similar to the method employed by
Guruswami and Xing (STOC 2013). However, our approach diverges by leveraging
the idea of {\it permuted product codes}, thereby simplifying the construction
by avoiding the need of {\it subspace designs}.
Specifically, the codes are constructed by initially forming the tensor
product of two RS codes with carefully selected evaluation sets, followed by
specific cyclic shifts to the codeword rows. This process results in each
codeword column being treated as an individual coordinate, reminiscent of prior
capacity-achieving codes, such as folded RS codes and univariate multiplicity
codes. This construction is easily shown to be a subcode of an interleaved RS
code, equivalently, an RS code evaluated on a subfield.
Alternatively, the codes can be constructed by the evaluation of bivariate
polynomials over orbits generated by \emph{two} affine transformations with
coprime orders, extending the earlier use of a single affine transformation in
folded RS codes and the recent affine folded RS codes introduced by Bhandari
{\it et al.} (IEEE T-IT, Feb.~2024). While our codes require large, yet
constant characteristic, the two affine transformations facilitate achieving
code length equal to the field size, without the restriction of the field being
prime, contrasting with univariate multiplicity codes.Comment: 20 page
Fractional repetition codes with flexible repair from combinatorial designs
Fractional repetition (FR) codes are a class of regenerating codes for
distributed storage systems with an exact (table-based) repair process that is
also uncoded, i.e., upon failure, a node is regenerated by simply downloading
packets from the surviving nodes. In our work, we present constructions of FR
codes based on Steiner systems and resolvable combinatorial designs such as
affine geometries, Hadamard designs and mutually orthogonal Latin squares. The
failure resilience of our codes can be varied in a simple manner. We construct
codes with normalized repair bandwidth () strictly larger than one;
these cannot be obtained trivially from codes with . Furthermore, we
present the Kronecker product technique for generating new codes from existing
ones and elaborate on their properties. FR codes with locality are those where
the repair degree is smaller than the number of nodes contacted for
reconstructing the stored file. For these codes we establish a tradeoff between
the local repair property and failure resilience and construct codes that meet
this tradeoff. Much of prior work only provided lower bounds on the FR code
rate. In our work, for most of our constructions we determine the code rate for
certain parameter ranges.Comment: 27 pages in IEEE two-column format. IEEE Transactions on Information
Theory (to appear
Subspace Evasive Sets
In this work we describe an explicit, simple, construction of large subsets
of F^n, where F is a finite field, that have small intersection with every
k-dimensional affine subspace. Interest in the explicit construction of such
sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl
(2004) who showed how such constructions over the binary field can be used to
construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that,
over large finite fields (of size polynomial in n), subspace evasive sets can
be used to obtain explicit list-decodable codes with optimal rate and constant
list-size. In this work we construct subspace evasive sets over large fields
and use them to reduce the list size of folded Reed-Solomon codes form poly(n)
to a constant.Comment: 16 page
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
Stabilizer quantum codes from -affine variety codes and a new Steane-like enlargement
New stabilizer codes with parameters better than the ones available in the
literature are provided in this work, in particular quantum codes with
parameters and that are records.
These codes are constructed with a new generalization of the Steane's
enlargement procedure and by considering orthogonal subfield-subcodes --with
respect to the Euclidean and Hermitian inner product-- of a new family of
linear codes, the -affine variety codes
- …