53,980 research outputs found
Generalized weights: an anticode approach
In this paper we study generalized weights as an algebraic invariant of a
code. We first describe anticodes in the Hamming and in the rank metric,
proving in particular that optimal anticodes in the rank metric coincide with
Frobenius-closed spaces. Then we characterize both generalized Hamming and rank
weights of a code in terms of the intersection of the code with optimal
anticodes in the respective metrics. Inspired by this description, we propose a
new algebraic invariant, which we call "Delsarte generalized weights", for
Delsarte rank-metric codes based on optimal anticodes of matrices. We show that
our invariant refines the generalized rank weights for Gabidulin codes proposed
by Kurihara, Matsumoto and Uyematsu, and establish a series of properties of
Delsarte generalized weights. In particular, we characterize Delsarte optimal
codes and anticodes in terms of their generalized weights. We also present a
duality theory for the new algebraic invariant, proving that the Delsarte
generalized weights of a code completely determine the Delsarte generalized
weights of the dual code. Our results extend the theory of generalized rank
weights for Gabidulin codes. Finally, we prove the analogue for Gabidulin codes
of a theorem of Wei, proving that their generalized rank weights characterize
the worst-case security drops of a Gabidulin rank-metric code
New Explicit Good Linear Sum-Rank-Metric Codes
Sum-rank-metric codes have wide applications in universal error correction
and security in multishot network, space-time coding and construction of
partial-MDS codes for repair in distributed storage. Fundamental properties of
sum-rank-metric codes have been studied and some explicit or probabilistic
constructions of good sum-rank-metric codes have been proposed. In this paper
we propose three simple constructions of explicit linear sum-rank-metric codes.
In finite length regime, numerous good linear sum-rank-metric codes from our
construction are given. Most of them have better parameters than previous
constructed sum-rank-metric codes. For example a lot of small block size better
linear sum-rank-metric codes over of the matrix size
are constructed for . Asymptotically our constructed sum-rank-metric
codes are closing to the Gilbert-Varshamov-like bound on sum-rank-metric codes
for some parameters. Finally we construct a linear MSRD code over an arbitrary
finite field with various matrix sizes
satisfying , , for any
given minimum sum-rank distance. There is no restriction on the block lengths
and parameters of these linear MSRD codes from the sizes
of the fields .Comment: 32 pages, revised version, merged with arXiv:2206.0233
Densities of Codes of Various Linearity Degrees in Translation-Invariant Metric Spaces
We investigate the asymptotic density of error-correcting codes with good
distance properties and prescribed linearity degree, including sublinear and
nonlinear codes. We focus on the general setting of finite
translation-invariant metric spaces, and then specialize our results to the
Hamming metric, to the rank metric, and to the sum-rank metric. Our results
show that the asymptotic density of codes heavily depends on the imposed
linearity degree and the chosen metric
Fundamental Properties of Sum-Rank Metric Codes
This paper investigates the theory of sum-rank metric codes for which the
individual matrix blocks may have different sizes. Various bounds on the
cardinality of a code are derived, along with their asymptotic extensions. The
duality theory of sum-rank metric codes is also explored, showing that MSRD
codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all
matrix blocks have the same number of columns. In the latter case, duality
considerations lead to an upper bound on the number of blocks for MSRD codes.
The paper also contains various constructions of sum-rank metric codes for
variable block sizes, illustrating the possible behaviours of these objects
with respect to bounds, existence, and duality properties
Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations
This paper establishes information-theoretic limits in estimating a finite
field low-rank matrix given random linear measurements of it. These linear
measurements are obtained by taking inner products of the low-rank matrix with
random sensing matrices. Necessary and sufficient conditions on the number of
measurements required are provided. It is shown that these conditions are sharp
and the minimum-rank decoder is asymptotically optimal. The reliability
function of this decoder is also derived by appealing to de Caen's lower bound
on the probability of a union. The sufficient condition also holds when the
sensing matrices are sparse - a scenario that may be amenable to efficient
decoding. More precisely, it is shown that if the n\times n-sensing matrices
contain, on average, \Omega(nlog n) entries, the number of measurements
required is the same as that when the sensing matrices are dense and contain
entries drawn uniformly at random from the field. Analogies are drawn between
the above results and rank-metric codes in the coding theory literature. In
fact, we are also strongly motivated by understanding when minimum rank
distance decoding of random rank-metric codes succeeds. To this end, we derive
distance properties of equiprobable and sparse rank-metric codes. These
distance properties provide a precise geometric interpretation of the fact that
the sparse ensemble requires as few measurements as the dense one. Finally, we
provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at
IEEE International Symposium on Information Theory (ISIT) 201
Correcting Charge-Constrained Errors in the Rank-Modulation Scheme
We investigate error-correcting codes for a the
rank-modulation scheme with an application to flash memory
devices. In this scheme, a set of n cells stores information in the
permutation induced by the different charge levels of the individual
cells. The resulting scheme eliminates the need for discrete
cell levels, overcomes overshoot errors when programming cells (a
serious problem that reduces the writing speed), and mitigates the
problem of asymmetric errors. In this paper, we study the properties
of error-correcting codes for charge-constrained errors in the
rank-modulation scheme. In this error model the number of errors
corresponds to the minimal number of adjacent transpositions required
to change a given stored permutation to another erroneous
one—a distance measure known as Kendall’s τ-distance.We show
bounds on the size of such codes, and use metric-embedding techniques
to give constructions which translate a wealth of knowledge
of codes in the Lee metric to codes over permutations in Kendall’s
τ-metric. Specifically, the one-error-correcting codes we construct
are at least half the ball-packing upper bound
Partitions of Matrix Spaces With an Application to -Rook Polynomials
We study the row-space partition and the pivot partition on the matrix space
. We show that both these partitions are reflexive
and that the row-space partition is self-dual. Moreover, using various
combinatorial methods, we explicitly compute the Krawtchouk coefficients
associated with these partitions. This establishes MacWilliams-type identities
for the row-space and pivot enumerators of linear rank-metric codes. We then
generalize the Singleton-like bound for rank-metric codes, and introduce two
new concepts of code extremality. Both of them generalize the notion of MRD
codes and are preserved by trace-duality. Moreover, codes that are extremal
according to either notion satisfy strong rigidity properties analogous to
those of MRD codes. As an application of our results to combinatorics, we give
closed formulas for the -rook polynomials associated with Ferrers diagram
boards. Moreover, we exploit connections between matrices over finite fields
and rook placements to prove that the number of matrices of rank over
supported on a Ferrers diagram is a polynomial in , whose
degree is strictly increasing in . Finally, we investigate the natural
analogues of the MacWilliams Extension Theorem for the rank, the row-space, and
the pivot partitions
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