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• W3048049228 abstract "A locally decodable code (LDC) maps K source symbols, each of size L <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>w</sub> bits, to M coded symbols, each of size L <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>x</sub> bits, such that each source symbol can be decoded from N ≤ M coded symbols. A perfectly smooth LDC further requires that each coded symbol is uniformly accessed when we decode any one of the messages. The ratio L <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>w</sub> /L <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>x</sub> is called the symbol rate of an LDC. The highest possible symbol rate for a class of LDCs is called the capacity of that class. It is shown that given K, N, the maximum value of capacity of perfectly smooth LDCs, maximized over all code lengths M, is C* = N(1 + 1/N 1/N <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sup> + · · · 1/N <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>K-1</sup> ) <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>-1</sup> . Furthermore, given K, N, the minimum code length M for which the capacity of a perfectly smooth LDC is C* is shown to be M = N <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>K</sup> . Both of these results generalize to a broader class of LDCs, called universal LDCs. The results are then translated into the context of PIR <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>max</sub> , i.e., Private Information Retrieval subject to maximum (rather than average) download cost metric. It is shown that the minimum upload cost of capacity achieving PIRmax schemes is (K - 1) log N. The results also generalize to a variation of the PIR problem, known as Repudiative Information Retrieval (RIR)." @default.
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• W3048049228 date "2020-10-01" @default.
• W3048049228 modified "2023-10-16" @default.
• W3048049228 title "On the Capacity of Locally Decodable Codes" @default.
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• W3048049228 doi "https://doi.org/10.1109/tit.2020.3014973" @default.
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