23 research outputs found
Further Progress on the GM-MDS Conjecture for Reed-Solomon Codes
Designing good error correcting codes whose generator matrix has a support
constraint, i.e., one for which only certain entries of the generator matrix
are allowed to be non-zero, has found many recent applications, including in
distributed coding and storage, multiple access networks, and weakly secure
data exchange. The dual problem, where the parity check matrix has a support
constraint, comes up in the design of locally repairable codes. The central
problem here is to design codes with the largest possible minimum distance,
subject to the given support constraint on the generator matrix. An upper bound
on the minimum distance can be obtained through a set of singleton bounds,
which can be alternatively thought of as a cut-set bound. Furthermore, it is
well known that, if the field size is large enough, any random generator matrix
obeying the support constraint will achieve the maximum minimum distance with
high probability. Since random codes are not easy to decode, structured codes
with efficient decoders, e.g., Reed-Solomon codes, are much more desirable. The
GM-MDS conjecture of Dau et al states that the maximum minimum distance over
all codes satisfying the generator matrix support constraint can be obtained by
a Reed Solomon code. If true, this would have significant consequences. The
conjecture has been proven for several special case: when the dimension of the
code k is less than or equal to five, when the number of distinct support sets
on the rows of the generator matrix m, say, is less than or equal to three, or
when the generator matrix is sparsest and balanced. In this paper, we report on
further progress on the GM-MDS conjecture. In particular, we show that the
conjecture is true for all m less than equal to six. This generalizes all
previous known results (except for the sparsest and balanced case, which is a
very special support constraint).Comment: Submitted to ISIT 201
Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero
Gabidulin codes over fields of characteristic zero were recently constructed by Augot et al., whenever the Galois group of the underlying field extension is cyclic. In parallel, the interest in sparse generator matrices of Reed–Solomon and Gabidulin codes has increased lately, due to applications in distributed computations. In particular, a certain condition pertaining to the intersection of zero entries at different rows, was shown to be necessary and sufficient for the existence of the sparsest possible generator matrix of Gabidulin codes over finite fields. In this paper we complete the picture by showing that the same condition is also necessary and sufficient for Gabidulin codes over fields of characteristic zero.Our proof builds upon and extends tools from the finite-field case, combines them with a variant of the Schwartz–Zippel lemma over automorphisms, and provides a simple randomized construction algorithm whose probability of success can be arbitrarily close to one. In addition, potential applications for low-rank matrix recovery are discussed
MDS matrices over small fields: A proof of the GM-MDS conjecture
An MDS matrix is a matrix whose minors all have full rank. A question arising
in coding theory is what zero patterns can MDS matrices have. There is a
natural combinatorial characterization (called the MDS condition) which is
necessary over any field, as well as sufficient over very large fields by a
probabilistic argument.
Dau et al. (ISIT 2014) conjectured that the MDS condition is sufficient over
small fields as well, where the construction of the matrix is algebraic instead
of probabilistic. This is known as the GM-MDS conjecture. Concretely, if a zero pattern satisfies the MDS condition, then they conjecture that
there exists an MDS matrix with this zero pattern over any field of size
. In recent years, this conjecture was proven in
several special cases. In this work, we resolve the conjecture
Weakly Secure MDS Codes for Simple Multiple Access Networks
We consider a simple multiple access network (SMAN), where sources of
unit rates transmit their data to a common sink via relays. Each relay is
connected to the sink and to certain sources. A coding scheme (for the relays)
is weakly secure if a passive adversary who eavesdrops on less than
relay-sink links cannot reconstruct the data from each source. We show that
there exists a weakly secure maximum distance separable (MDS) coding scheme for
the relays if and only if every subset of relays must be collectively
connected to at least sources, for all . Moreover, we
prove that this condition can be verified in polynomial time in and .
Finally, given a SMAN satisfying the aforementioned condition, we provide
another polynomial time algorithm to trim the network until it has a sparsest
set of source-relay links that still supports a weakly secure MDS coding
scheme.Comment: Accepted at ISIT'1
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
Cooperative Data Exchange with Unreliable Clients
Consider a set of clients in a broadcast network, each of which holds a
subset of packets in the ground set X. In the (coded) cooperative data exchange
problem, the clients need to recover all packets in X by exchanging coded
packets over a lossless broadcast channel. Several previous works analyzed this
problem under the assumption that each client initially holds a random subset
of packets in X. In this paper we consider a generalization of this problem for
settings in which an unknown (but of a certain size) subset of clients are
unreliable and their packet transmissions are subject to arbitrary erasures.
For the special case of one unreliable client, we derive a closed-form
expression for the minimum number of transmissions required for each reliable
client to obtain all packets held by other reliable clients (with probability
approaching 1 as the number of packets tends to infinity). Furthermore, for the
cases with more than one unreliable client, we provide an approximation
solution in which the number of transmissions per packet is within an
arbitrarily small additive factor from the value of the optimal solution.Comment: 8 pages; in Proc. 53rd Annual Allerton Conference on Communication,
Control, and Computing (Allerton 2015
Coded Cooperative Data Exchange for a Secret Key
We consider a coded cooperative data exchange problem with the goal of
generating a secret key. Specifically, we investigate the number of public
transmissions required for a set of clients to agree on a secret key with
probability one, subject to the constraint that it remains private from an
eavesdropper.
Although the problems are closely related, we prove that secret key
generation with fewest number of linear transmissions is NP-hard, while it is
known that the analogous problem in traditional cooperative data exchange can
be solved in polynomial time. In doing this, we completely characterize the
best possible performance of linear coding schemes, and also prove that linear
codes can be strictly suboptimal. Finally, we extend the single-key results to
characterize the minimum number of public transmissions required to generate a
desired integer number of statistically independent secret keys.Comment: Full version of a paper that appeared at ISIT 2014. 19 pages, 2
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Optimum Linear Codes with Support Constraints over Small Fields
The problem of designing a linear code with the largest possible minimum distance, subject to support constraints on the generator matrix, has recently found several applications. These include multiple access networks [3], [5] as well as weakly secure data exchange [4], [8]. A simple upper bound on the maximum minimum distance can be obtained from a sequence of Singleton bounds (see (3) below) and can further be achieved by randomly choosing the nonzero elements of the generator matrix from a field of a large enough size
Optimum Linear Codes with Support Constraints over Small Fields
We consider the problem of designing optimal linear codes (in terms of having
the largest minimum distance) subject to a support constraint on the generator
matrix. We show that the largest minimum distance can be achieved by a subcode
of a Reed-Solomon code of small field size. As a by-product of this result, we
settle the GM-MDS conjecture of Dau et. al. in the affirmative