2,299 research outputs found
On the Existence of MDS Codes Over Small Fields With Constrained Generator Matrices
We study the existence over small fields of Maximum Distance Separable (MDS)
codes with generator matrices having specified supports (i.e. having specified
locations of zero entries). This problem unifies and simplifies the problems
posed in recent works of Yan and Sprintson (NetCod'13) on weakly secure
cooperative data exchange, of Halbawi et al. (arxiv'13) on distributed
Reed-Solomon codes for simple multiple access networks, and of Dau et al.
(ISIT'13) on MDS codes with balanced and sparse generator matrices. We
conjecture that there exist such MDS codes as long as , if the specified supports of the generator matrices satisfy the so-called
MDS condition, which can be verified in polynomial time. We propose a
combinatorial approach to tackle the conjecture, and prove that the conjecture
holds for a special case when the sets of zero coordinates of rows of the
generator matrix share with each other (pairwise) at most one common element.
Based on our numerical result, the conjecture is also verified for all . Our approach is based on a novel generalization of the well-known Hall's
marriage theorem, which allows (overlapping) multiple representatives instead
of a single representative for each subset.Comment: 8 page
Generalized GM-MDS: Polynomial Codes are Higher Order MDS
The GM-MDS theorem, conjectured by Dau-Song-Dong-Yuen and proved by Lovett
and Yildiz-Hassibi, shows that the generator matrices of Reed-Solomon codes can
attain every possible configuration of zeros for an MDS code. The recently
emerging theory of higher order MDS codes has connected the GM-MDS theorem to
other important properties of Reed-Solomon codes, including showing that
Reed-Solomon codes can achieve list decoding capacity, even over fields of size
linear in the message length.
A few works have extended the GM-MDS theorem to other families of codes,
including Gabidulin and skew polynomial codes. In this paper, we generalize all
these previous results by showing that the GM-MDS theorem applies to any
\emph{polynomial code}, i.e., a code where the columns of the generator matrix
are obtained by evaluating linearly independent polynomials at different
points. We also show that the GM-MDS theorem applies to dual codes of such
polynomial codes, which is non-trivial since the dual of a polynomial code may
not be a polynomial code. More generally, we show that GM-MDS theorem also
holds for algebraic codes (and their duals) where columns of the generator
matrix are chosen to be points on some irreducible variety which is not
contained in a hyperplane through the origin. Our generalization has
applications to constructing capacity-achieving list-decodable codes as shown
in a follow-up work by Brakensiek-Dhar-Gopi-Zhang, where it is proved that
randomly punctured algebraic-geometric (AG) codes achieve list-decoding
capacity over constant-sized fields.Comment: 34 page
AG codes achieve list decoding capacity over contant-sized fields
The recently-emerging field of higher order MDS codes has sought to unify a
number of concepts in coding theory. Such areas captured by higher order MDS
codes include maximally recoverable (MR) tensor codes, codes with optimal
list-decoding guarantees, and codes with constrained generator matrices (as in
the GM-MDS theorem).
By proving these equivalences, Brakensiek-Gopi-Makam showed the existence of
optimally list-decodable Reed-Solomon codes over exponential sized fields.
Building on this, recent breakthroughs by Guo-Zhang and Alrabiah-Guruswami-Li
have shown that randomly punctured Reed-Solomon codes achieve list-decoding
capacity (which is a relaxation of optimal list-decodability) over linear size
fields. We extend these works by developing a formal theory of relaxed higher
order MDS codes. In particular, we show that there are two inequivalent
relaxations which we call lower and upper relaxations. The lower relaxation is
equivalent to relaxed optimal list-decodable codes and the upper relaxation is
equivalent to relaxed MR tensor codes with a single parity check per column.
We then generalize the techniques of GZ and AGL to show that both these
relaxations can be constructed over constant size fields by randomly puncturing
suitable algebraic-geometric codes. For this, we crucially use the generalized
GM-MDS theorem for polynomial codes recently proved by Brakensiek-Dhar-Gopi. We
obtain the following corollaries from our main result. First, randomly
punctured AG codes of rate achieve list-decoding capacity with list size
and field size . Prior to this work, AG
codes were not even known to achieve list-decoding capacity. Second, by
randomly puncturing AG codes, we can construct relaxed MR tensor codes with a
single parity check per column over constant-sized fields, whereas
(non-relaxed) MR tensor codes require exponential field size.Comment: 38 page
Systematic MDS erasure codes based on vandermonde matrices
An increasing number of applications in computer communications uses erasure codes to cope with packet losses. Systematic maximum-distance separable (MDS) codes are often the best adapted codes. This letter introduces new systematic MDS erasure codes constructed from two Vandermonde matrices. These codes have lower coding and decoding complexities than the others systematic MDS erasure codes
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