11 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
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
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
Reed-Solomon codes over small fields with constrained generator matrices
We give constructions of some special cases of Reed-Solomon codes
over finite fields of size at least and whose generator matrices have
constrained support. Furthermore, we consider a generalisation of the GM-MDS
conjecture proposed by Lovett in 2018. We show that Lovett's conjecture is
false in general and we specify when the conjecture is true.Comment: 21 page
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 nonzero, 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. 1. 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)
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
Gabidulin Codes with Support Constrained Generator Matrices
Gabidulin codes are the first general construction of linear codes that are
maximum rank distant (MRD). They have found applications in linear network
coding, for example, when the transmitter and receiver are oblivious to the
inner workings and topology of the network (the so-called incoherent regime).
The reason is that Gabidulin codes can be used to map information to linear
subspaces, which in the absence of errors cannot be altered by linear
operations, and in the presence of errors can be corrected if the subspace is
perturbed by a small rank. Furthermore, in distributed coding and distributed
systems, one is led to the design of error correcting codes whose generator
matrix must satisfy a given support constraint. In this paper, we give
necessary and sufficient conditions on the support of the generator matrix that
guarantees the existence of Gabidulin codes and general MRD codes. When the
rate of the code is not very high, this is achieved with the same field size
necessary for Gabidulin codes with no support constraint. When these conditions
are not satisfied, we characterize the largest possible rank distance under the
support constraints and show that they can be achieved by subcodes of Gabidulin
codes. The necessary and sufficient conditions are identical to those that
appear for MDS codes which were recently proven by Yildiz et al. and Lovett in
the context of settling the GM-MDS conjecture
Generalized Reed-Solomon Codes with Sparsest and Balanced Generator Matrices
We prove that for any positive integers and such that , there exists an generalized Reed-Solomon (GRS) code that
has a sparsest and balanced generator matrix (SBGM) over any finite field of
size , where sparsest means that
each row of the generator matrix has the least possible number of nonzeros,
while balanced means that the number of nonzeros in any two columns differ by
at most one. Previous work by Dau et al (ISIT'13) showed that there always
exists an MDS code that has an SBGM over any finite field of size , and Halbawi et al (ISIT'16, ITW'16) showed that there exists
a cyclic Reed-Solomon code (i.e., ) with an SBGM for any prime power
. Hence, this work extends both of the previous results
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 nonzero, 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. 1. 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)