525 research outputs found
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Simplicity conditions for binary orthogonal arrays
It is known that correlation-immune (CI) Boolean functions used in the framework of side channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays, when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of rows in a simple orthogonal array. In this paper, we use Rao's Bound to give a sufficient condition on the number of rows, for a binary orthogonal array (OA) to be simple. We apply this result for determining the minimum number of rows in all simple binary orthogonal arrays of strengths 2 and 3; we show that this minimum is the same in such case as for all OA, and we extend this observation to some OA of strengths 4 and 5. This allows us to reply positively, in the case of strengths 2 and 3, to a question raised by the first author and X. Chen on the monotonicity of the minimum Hamming weight of 2-CI Boolean functions, and to partially reply positively to the same question in the case of strengths 4 and 5
Numerical and analytical bounds on threshold error rates for hypergraph-product codes
We study analytically and numerically decoding properties of finite rate
hypergraph-product quantum LDPC codes obtained from random (3,4)-regular
Gallager codes, with a simple model of independent X and Z errors. Several
non-trival lower and upper bounds for the decodable region are constructed
analytically by analyzing the properties of the homological difference, equal
minus the logarithm of the maximum-likelihood decoding probability for a given
syndrome. Numerical results include an upper bound for the decodable region
from specific heat calculations in associated Ising models, and a minimum
weight decoding threshold of approximately 7%.Comment: 14 pages, 5 figure
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Algorithm Based Fault Tolerance in Massively Parallel Systems
An A complex computer system consists of billions of transistors, miles of wires, and many interactions with an unpredictable environment. Correct results must be produced despite faults that dynamically occur in some of these components. Many techniques have been developed for fault tolerant computation. General purpose methods are independent of the application, yet incur an overhead cost which may be unacceptable for massively parallel systems. Algorithm-specific methods, which can operate at lower cost, are a developing alternative [1, 72]. This paper first reviews the general-purpose approach and then focuses on the algorithm-specific method, with an eye toward massively parallel processors. Algorithm-based fault tolerance has the attraction of low overhead; furthermore it addresses both the detection and also the correction problems. The principle is to build low-cost checking and correcting mechanism based exclusively on the redundancies inherent in the system
ΠΠ΅Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠ΅ ΡΠ»ΡΡΠ°ΠΉΠ½ΠΎΠ΅ ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅ Π² ΡΠΈΡΡΠ΅ΠΌΠ°Ρ ΠΏΠ΅ΡΠ΅Π΄Π°ΡΠΈ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΈ ΠΏΠΎ ΠΊΠ°Π½Π°Π»Ρ ΡΠ²ΡΠ·ΠΈ Ρ ΠΎΡΠ²ΠΎΠ΄ΠΎΠΌ
Non-asymptotically bounds for the probability of optimal messages decoding in the wiretap channel of a information transmission system with random coding by a arbitrary resilient function over an Abelian group, are obtained. Algorithms of random messages coding and decoding in main channel which use non-linear systhematic codes, are described. An example of the information transmission system with random coding by Preparata codes is considered.ΠΠΎΠ»ΡΡΠ΅Π½Ρ Π½Π΅Π°ΡΠΈΠΌΠΏΡΠΎΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ Π³ΡΠ°Π½ΠΈΡΡ Π²Π΅ΡΠΎΡΡΠ½ΠΎΡΡΠΈ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠΈΠ΅ΠΌΠ° ΡΠΎΠΎΠ±ΡΠ΅Π½ΠΈΠΉ Π² ΠΎΡΠ²ΠΎΠ΄Π½ΠΎΠΌ ΠΊΠ°Π½Π°Π»Π΅ ΡΠΈΡΡΠ΅ΠΌΡ ΠΏΠ΅ΡΠ΅Π΄Π°ΡΠΈ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΈ ΡΠΎ ΡΠ»ΡΡΠ°ΠΉΠ½ΡΠΌ ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅ΠΌ, ΠΏΠΎΡΡΡΠΎΠ΅Π½Π½ΠΎΠΉ Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ»ΡΠ½ΠΎΠΉ ΡΡΡΠΎΠΉΡΠΈΠ²ΠΎΠΉ ΡΡΠ½ΠΊΡΠΈΠΈ Π½Π°Π΄ ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠΉ Π°Π±Π΅Π»Π΅Π²ΠΎΠΉ Π³ΡΡΠΏΠΏΠΎΠΉ. ΠΠΏΠΈΡΠ°Π½Ρ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ ΡΠ»ΡΡΠ°ΠΉΠ½ΠΎΠ³ΠΎ ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΈ Π΄Π΅ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΡΠΎΠΎΠ±ΡΠ΅Π½ΠΈΠΉ Π² ΠΎΡΠ½ΠΎΠ²Π½ΠΎΠΌ ΠΊΠ°Π½Π°Π»Π΅ Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠ΅ΠΌ Π½Π΅Π»ΠΈΠ½Π΅ΠΉΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΊΠΎΠ΄ΠΎΠ². Π Π°ΡΡΠΌΠΎΡΡΠ΅Π½ ΠΏΡΠΈΠΌΠ΅Ρ ΡΠΈΡΡΠ΅ΠΌΡ ΠΏΠ΅ΡΠ΅Π΄Π°ΡΠΈ ΡΠΎ ΡΠ»ΡΡΠ°ΠΉΠ½ΡΠΌ ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅ΠΌ, ΠΏΠΎΡΡΡΠΎΠ΅Π½Π½ΠΎΠΉ Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΊΠΎΠ΄ΠΎΠ² ΠΡΠ΅ΠΏΠ°ΡΠ°ΡΡ
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