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Subspace Evasive Sets
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 139 (2011)
, 2011
"... In this work we describe an explicit, simple, construction of large subsets of F n, where F is a finite field, that have small intersection with every kdimensional affine subspace. Interest in the explicit construction of such sets, termed subspaceevasive sets, started in the work of Pudlák and Rö ..."
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In this work we describe an explicit, simple, construction of large subsets of F n, where F is a finite field, that have small intersection with every kdimensional affine subspace. Interest in the explicit construction of such sets, termed subspaceevasive sets, started in the work of Pudlák and Rödl [PR04] who showed how such constructions over the binary field can be used to construct explicit Ramsey graphs. More recently, Guruswami [Gur11] showed that, over large finite fields (of size polynomial in n), subspace evasive sets can be used to obtain explicit listdecodable codes with optimal rate and constant listsize. In this work we construct subspace evasive sets over large fields and use them, as described in [Gur11], to reduce the list size of folded ReedSolomon codes form poly(n) to a constant.
List decoding subspace codes from insertions and deletions, arxiv.org/abs/1202.0535
, 2012
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Combinatorial Limitations of Averageradius List Decoding ⋆
"... Abstract. We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/γ)) and lower bound (of Ωp(log(1/γ))) for the listsize needed to list decode up to error fraction p with rate γ away from capacity, i.e., 1−h(p)−γ (here p ∈ (0, ..."
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Abstract. We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/γ)) and lower bound (of Ωp(log(1/γ))) for the listsize needed to list decode up to error fraction p with rate γ away from capacity, i.e., 1−h(p)−γ (here p ∈ (0, 1 2) and γ> 0). Our main result is the following: We prove that in any binary code C ⊆ {0, 1} n of rate 1−h(p)−γ, there must exist a set L ⊂ C of Ωp(1 / √ γ) codewords such that the average distance of the points in L from their centroid is at most pn. In other words, there must exist Ωp(1 / √ γ) codewords with low “average radius. ” The standard notion of listdecoding corresponds to working with the maximum distance of a collection of codewords from a center instead of average distance. The averageradius form is in itself quite natural; for instance, the classical Johnson bound in fact implies averageradius listdecodability.
Group homomorphisms as error correcting codes
"... We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups G and H. We prove some general structural results on how the distance behaves with respect to natural group operations, such as passing to subgroups and quotients, and taking produ ..."
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We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups G and H. We prove some general structural results on how the distance behaves with respect to natural group operations, such as passing to subgroups and quotients, and taking products. Our main result is a general formula for the distance when G is solvable or H is nilpotent, in terms of the normal subgroup structure of G as well as the prime divisors of G and H. In particular, we show that in the above case, the distance is independent of the subgroup structure of H. We complement this by showing that, in general, the distance depends on the subgroup structure of H. 1
A General Construction for 1round δRMT and (0, δ)SMT
"... Abstract. In Secure Message Transmission (SMT) problem, a sender S is connected to a receiver R through N node disjoint bidirectional paths in the network, t of which are controlled by an adversary with unlimited computational power. S wants to send a message m to R in a reliable and private way. It ..."
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Abstract. In Secure Message Transmission (SMT) problem, a sender S is connected to a receiver R through N node disjoint bidirectional paths in the network, t of which are controlled by an adversary with unlimited computational power. S wants to send a message m to R in a reliable and private way. It is proved that SMT is possible if and only if N ≥ 2t+1. In Reliable Message Transmission (RMT) problem, the network setting is the same and the goal is to provide reliability for communication, only. In this paper we focus on 1round δRMT and (0, δ)SMT where the chance of protocol failure (receiver cannot decode the sent message) is at most δ, and in the case of SMT, privacy is perfect. We propose a new approach to the construction of 1round δRMT and (0, δ)SMT for all connectivities N ≥ 2t + 1, using list decodable codes and message authentication codes. Our concrete constructions use folded ReedSolomon codes and multireceiver message authentication codes. The protocols have optimal transmission rates and provide the highest reliability among all known comparable protocols. Important advantages of these constructions are, (i) they can be adapted to all connectivities, and (ii) have simple and direct security (privacy and reliability) proofs using properties of the underlying codes, and δ can be calculated from parameters of the underlying codes. We discuss our results in relation to previous work in this area and propose directions for future research. 1
Variety Evasive Sets
"... We give an explicit construction of a large subset S ⊂ F n, where F is a finite field, that has small intersection with any affine variety of fixed dimension and bounded degree. Our construction generalizes a recent result of Dvir and Lovett (STOC 2012) who considered varieties of degree one (that i ..."
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We give an explicit construction of a large subset S ⊂ F n, where F is a finite field, that has small intersection with any affine variety of fixed dimension and bounded degree. Our construction generalizes a recent result of Dvir and Lovett (STOC 2012) who considered varieties of degree one (that is, affine subspaces). 1
Steganography in Audio Files by Entropy using FEC as Reed –Solomon of VOIP Streams
"... In this paper introduce a novel technique to identify the voice (active frames) and silent regions (inactive frames) of a speech stream very much suitable for VoIP calls. Thus here the proposed a better voice activity detection based on the entropy algorithm. Highcapacity steganography algorithm fo ..."
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In this paper introduce a novel technique to identify the voice (active frames) and silent regions (inactive frames) of a speech stream very much suitable for VoIP calls. Thus here the proposed a better voice activity detection based on the entropy algorithm. Highcapacity steganography algorithm for embedding data in the inactive frames.Then inactive frames are encoded by G.723.1 source codec, which is used extensively in Voice over Internet Protocol (VoIP).As the data embedding capacity is very high on inactive frames of the audio signals than in the active frames. Entropy based Voice Activity Detection algorithms for VoIP applications can save bandwidth by filtering the frames that do not contain speech.On evaluating the proposed approach with the existing methods, our approach yield a better saving in bandwidth, yet maintaining high capacity of data embedding. yet maintaining good quality of the speech streams and then finally using forward error correcting code as ReedSolomon codes. It can be used as encoder and decoder. By using ReedSolomon code, data losses occur in the transmission can be detected and recovered by adding extra information (redundancy) to the original data. Key words
SPECIAL ISSUE: APPROXRANDOM 2012 Extractors for Polynomial Sources over Fields of Constant Order and Small Characteristic ∗
, 2012
"... Abstract: A polynomial source of randomness over Fn q is a random variable X = f (Z) where f is a polynomial map and Z is a random variable distributed uniformly over Fr q for some integer r. The three main parameters of interest associated with a polynomial source are the order q of the field, the ..."
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Abstract: A polynomial source of randomness over Fn q is a random variable X = f (Z) where f is a polynomial map and Z is a random variable distributed uniformly over Fr q for some integer r. The three main parameters of interest associated with a polynomial source are the order q of the field, the (total) degree D of the map f, and the baseq logarithm of the size of the range of f over inputs in Fr q, denoted by k. For simplicity we call X a (q,D,k)source. Informally, an extractor for (q,D,k)sources is a function E: Fn q → {0,1} m such that the distribution of the random variable E(X) is close to uniform over {0,1} m for any (q,D,k)source X. Generally speaking, the problem of constructing extractors for such sources becomes harder as q and k decrease and as D increases. A rather large number of recent ∗A conference version of this paper appeared in the Proceedings of RANDOM 2012 [1]. † Supported by funding from the European Community’s Seventh Framework Programme (FP7/20072013) under grant agreement number 240258. ‡ Supported by funding from the European Community’s Seventh Framework Programme (FP7/20072013) under grant agreement number 240258.