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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.
Combinatorial Limitations of Averageradius List Decoding
"... 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 ..."
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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
, 2015
"... 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 produc ..."
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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.
Decoding of RepeatedRoot Cyclic Codes up to New Bounds on Their Minimum Distance
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Dimension Expanders via Rank Condensers
, 2014
"... An emerging theory of “linearalgebraic pseudorandomness” aims to understand the linearalgebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such al ..."
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An emerging theory of “linearalgebraic pseudorandomness” aims to understand the linearalgebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such algebraic objects such as subspace designs, dimension expanders, seeded rank condensers, twosource rank condensers, and rankmetric codes. In particular, with the recent construction of nearoptimal subspace designs by Guruswami and Kopparty [GK13] as a starting point, we construct good (seeded) rank condensers (both lossless and lossy versions), which are a small collection of linear maps Fn → Ft for t n such that for every subset of Fn of small rank, its rank is preserved (up to a constant factor in the lossy case) by at least one of the maps. We then compose a tensoring operation with our lossy rank condenser to construct constantdegree dimension expanders over polynomially large fields. That is, we give O(1) explicit linear maps Ai: Fn→ Fn such that for any subspace V ⊆ Fn of dimension at most n/2, dim(∑i Ai(V))> (1+Ω(1))dim(V). Previous constructions of such constantdegree dimension expanders were based on Kazhdan’s property T (for the case when F has characteristic zero) or monotone expanders (for every field F); in either case
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