6 research outputs found
Asymptotically good binary linear codes with asymptotically good self-intersection spans
If C is a binary linear code, let C^2 be the linear code spanned by
intersections of pairs of codewords of C. We construct an asymptotically good
family of binary linear codes such that, for C ranging in this family, the C^2
also form an asymptotically good family. For this we use algebraic-geometry
codes, concatenation, and a fair amount of bilinear algebra.
More precisely, the two main ingredients used in our construction are, first,
a description of the symmetric square of an odd degree extension field in terms
only of field operations of small degree, and second, a recent result of
Garcia-Stichtenoth-Bassa-Beelen on the number of points of curves on such an
odd degree extension field.Comment: 18 pages; v2->v3: expanded introduction and bibliography + various
minor change
An upper bound of Singleton type for componentwise products of linear codes
We give an upper bound that relates the minimum weight of a nonzero
componentwise product of codewords from some given number of linear codes, with
the dimensions of these codes. Its shape is a direct generalization of the
classical Singleton bound.Comment: 9 pages; major improvements in v3: now works for an arbitrary number
of codes, and the low-weight codeword can be taken in product form; submitted
to IEEE Trans. Inform. Theor
On squares of cyclic codes
The square of a linear error correcting code is the linear code
spanned by the component-wise products of every pair of (non-necessarily
distinct) words in . Squares of codes have gained attention for several
applications mainly in the area of cryptography, and typically in those
applications one is concerned about some of the parameters (dimension, minimum
distance) of both and . In this paper, motivated mostly by the
study of this problem in the case of linear codes defined over the binary
field, squares of cyclic codes are considered. General results on the minimum
distance of the squares of cyclic codes are obtained and constructions of
cyclic codes with relatively large dimension of and minimum distance of
the square are discussed. In some cases, the constructions lead to
codes such that both and simultaneously have the largest
possible minimum distances for their length and dimensions.Comment: Accepted at IEEE Transactions on Information Theory. IEEE early
access version available at https://ieeexplore.ieee.org/document/8451926
Torsion Limits and Riemann-Roch Systems for Function Fields and Applications
The Ihara limit (or -constant) has been a central problem of study in
the asymptotic theory of global function fields (or equivalently, algebraic
curves over finite fields). It addresses global function fields with many
rational points and, so far, most applications of this theory do not require
additional properties. Motivated by recent applications, we require global
function fields with the additional property that their zero class divisor
groups contain at most a small number of -torsion points. We capture this by
the torsion limit, a new asymptotic quantity for global function fields. It
seems that it is even harder to determine values of this new quantity than the
Ihara constant. Nevertheless, some non-trivial lower- and upper bounds are
derived. Apart from this new asymptotic quantity and bounds on it, we also
introduce Riemann-Roch systems of equations. It turns out that this type of
equation system plays an important role in the study of several other problems
in areas such as coding theory, arithmetic secret sharing and multiplication
complexity of finite fields etc. Finally, we show how our new asymptotic
quantity, our bounds on it and Riemann-Roch systems can be used to improve
results in these areas.Comment: Accepted for publication in IEEE Transactions on Information Theory.
This is an extended version of our paper in Proceedings of 31st Annual IACR
CRYPTO, Santa Barbara, Ca., USA, 2011. The results in Sections 5 and 6 did
not appear in that paper. A first version of this paper has been widely
circulated since November 200
Bounded Indistinguishability and the Complexity of Recovering Secrets
Motivated by cryptographic applications, we study the notion of {\em bounded indistinguishability}, a natural relaxation of the well studied notion of bounded independence.
We say that two distributions and over are {\em -wise indistinguishable} if their projections to any symbols are identical. We say that a function f\colon \Sigma^n \to \zo is {\em \e-fooled by -wise indistinguishability} if cannot distinguish with
advantage \e between any two -wise indistinguishable distributions and over
.
We are interested in characterizing the class of functions that are fooled by -wise indistinguishability. While the case of -wise independence (corresponding to one of the distributions being uniform) is fairly well understood, the more general case remained unexplored.
When \Sigma = \zo, we observe that whether is fooled is closely related
to its approximate degree. For larger alphabets , we obtain several positive and negative
results. Our results imply the first efficient secret sharing schemes with a high secrecy threshold
in which the secret can be reconstructed in AC. More concretely, we show that for every
it is possible to share a secret among parties so that
any set of fewer than parties can learn nothing about the secret,
any set of at least parties can reconstruct the secret, and where
both the sharing and the reconstruction are done by constant-depth circuits
of size \poly(n). We present additional cryptographic applications of our results to low-complexity secret sharing, visual secret sharing, leakage-resilient cryptography, and protecting against ``selective failure\u27\u27 attacks
On products and powers of linear codes under componentwise multiplication
In this text we develop the formalism of products and powers of linear codes
under componentwise multiplication. As an expanded version of the author's talk
at AGCT-14, focus is put mostly on basic properties and descriptive statements
that could otherwise probably not fit in a regular research paper. On the other
hand, more advanced results and applications are only quickly mentioned with
references to the literature. We also point out a few open problems.
Our presentation alternates between two points of view, which the theory
intertwines in an essential way: that of combinatorial coding, and that of
algebraic geometry.
In appendices that can be read independently, we investigate topics in
multilinear algebra over finite fields, notably we establish a criterion for a
symmetric multilinear map to admit a symmetric algorithm, or equivalently, for
a symmetric tensor to decompose as a sum of elementary symmetric tensors.Comment: 75 pages; expanded version of a talk at AGCT-14 (Luminy), to appear
in vol. 637 of Contemporary Math., AMS, Apr. 2015; v3: minor typos corrected
in the final "open questions" sectio