
Density of Binary Disc Packings:Lower and Upper Bounds
We provide, for any r∈ (0,1), lower and upper bounds on the maximal dens...
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Combinatorial listdecoding of ReedSolomon codes beyond the Johnson radius
Listdecoding of ReedSolomon (RS) codes beyond the so called Johnson ra...
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On twofold packings of radius1 balls in Hamming graphs
A λfold rpacking in a Hamming metric space is a code C such that the r...
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Improved ListDecodability of Reed–Solomon Codes via Tree Packings
This paper shows that there exist Reed–Solomon (RS) codes, over large fi...
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List Decoding of Locally Repairable Codes
We show that locally repairable codes (LRCs) can be list decoded efficie...
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On Approximation, Bounding Exact Calculation of Block Error Probability for Random Codes
This paper presents a method to calculate the exact block error probabil...
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Consistent High Dimensional Rounding with Side Information
In standard rounding, we want to map each value X in a large continuous ...
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Bounds for Multiple Packing and ListDecoding Error Exponents
We revisit the problem of highdimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let N>0 and L∈ℤ_≥2. A multiple packing is a set 𝒞 of points in ℝ^n such that any point in ℝ^n lies in the intersection of at most L1 balls of radius √(nN) around points in 𝒞. We study the multiple packing problem for both bounded point sets whose points have norm at most √(nP) for some constant P>0 and unbounded point sets whose points are allowed to be anywhere in ℝ^n. Given a wellknown connection with coding theory, multiple packings can be viewed as the Euclidean analog of listdecodable codes, which are wellstudied for finite fields. In this paper, we derive various bounds on the largest possible density of a multiple packing in both bounded and unbounded settings. A related notion called averageradius multiple packing is also studied. Some of our lower bounds exactly pin down the asymptotics of certain ensembles of averageradius listdecodable codes, e.g., (expurgated) Gaussian codes and (expurgated) Poisson Point Processes. To this end, we apply tools from highdimensional geometry and large deviation theory. Some of our lower bounds on the optimal multiple packing density are the best known lower bounds. These bounds are obtained via a proxy known as error exponent. The latter quantity is the best exponent of the probability of listdecoding error when the code is corrupted by a Gaussian noise. We establish a curious inequality which relates the error exponent, a quantity of averagecase nature, to the listdecoding radius, a quantity of worstcase nature. We derive various bounds on the error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.
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