97 research outputs found
Weights of irreducible cyclic codes
With any fixed prime number p and positive integer N, not divisible by p, there is associated an infinite sequence of cyclic codes. In a previous article it was shown that a theorem of Davenport-Hasse reduces the calculation of the weight distributions for this whole sequence of codes to a single calculation (essentially that of calculating the weight distribution for the simplest code of the sequence). The primary object of this paper is the development of machinery which simplifies this remaining calculation. Detailed examples are given. In addition, tables are presented which essentially solve the weight distribution problem for all such binary codes with N < 100 and, when the block length is less than one million, give the complete weight enumerator
Spectral approach to linear programming bounds on codes
We give new proofs of asymptotic upper bounds of coding theory obtained
within the frame of Delsarte's linear programming method. The proofs rely on
the analysis of eigenvectors of some finite-dimensional operators related to
orthogonal polynomials. The examples of the method considered in the paper
include binary codes, binary constant-weight codes, spherical codes, and codes
in the projective spaces.Comment: 11 pages, submitte
Leakage-resilient coin tossing
Proceedings 25th International Symposium, DISC 2011, Rome, Italy, September 20-22, 2011.The ability to collectively toss a common coin among n parties
in the presence of faults is an important primitive in the arsenal of
randomized distributed protocols. In the case of dishonest majority, it
was shown to be impossible to achieve less than 1
r bias in O(r) rounds
(Cleve STOC ’86). In the case of honest majority, in contrast, unconditionally
secure O(1)-round protocols for generating common unbiased
coins follow from general completeness theorems on multi-party secure
protocols in the secure channels model (e.g., BGW, CCD STOC ’88).
However, in the O(1)-round protocols with honest majority, parties
generate and hold secret values which are assumed to be perfectly hidden
from malicious parties: an assumption which is crucial to proving the
resulting common coin is unbiased. This assumption unfortunately does
not seem to hold in practice, as attackers can launch side-channel attacks
on the local state of honest parties and leak information on their secrets.
In this work, we present an O(1)-round protocol for collectively generating
an unbiased common coin, in the presence of leakage on the local
state of the honest parties. We tolerate t ≤ ( 1
3
− )n computationallyunbounded
Byzantine faults and in addition a Ω(1)-fraction leakage on
each (honest) party’s secret state. Our results hold in the memory leakage
model (of Akavia, Goldwasser, Vaikuntanathan ’08) adapted to the
distributed setting.
Additional contributions of our work are the tools we introduce to
achieve the collective coin toss: a procedure for disjoint committee election,
and leakage-resilient verifiable secret sharing.National Defense Science and Engineering Graduate FellowshipNational Science Foundation (U.S.) (CCF-1018064
Quantum Stabilizer Codes and Classical Linear Codes
We show that within any quantum stabilizer code there lurks a classical
binary linear code with similar error-correcting capabilities, thereby
demonstrating new connections between quantum codes and classical codes. Using
this result -- which applies to degenerate as well as nondegenerate codes --
previously established necessary conditions for classical linear codes can be
easily translated into necessary conditions for quantum stabilizer codes.
Examples of specific consequences are: for a quantum channel subject to a
delta-fraction of errors, the best asymptotic capacity attainable by any
stabilizer code cannot exceed H(1/2 + sqrt(2*delta*(1-2*delta))); and, for the
depolarizing channel with fidelity parameter delta, the best asymptotic
capacity attainable by any stabilizer code cannot exceed 1-H(delta).Comment: 17 pages, ReVTeX, with two figure
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