Generalized Singleton Type Upper Bounds
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by
Hao Chen
2022
Abstract
In this paper we give upper bounds on the sizes of (d, L) list-decodable
codes in the Hamming metric space from various covering codes with the covering
radius d. When the list size L is 1, this gives many new Singleton type
upper bounds on the sizes of codes with a given minimum Hamming distance. These
upper bounds for codes are tighter than the Griesmer bound when the lengths of
codes are large. Some upper bounds on the lengths of general small Singleton
defect are given. An upper bound on the lengths of list-decodable codes
attaining the generalized Singleton bound is also presented. As an application
of our generalized Singleton type upper bounds on Hamming metric
error-correcting codes, the generalized Singleton type upper bounds on
insertion-deletion codes is given. Our this upper bound is much stronger than
the direct Singleton bound for insertion-deletion codes when the lengths are
large. We also give upper bounds on the lengths of small dimension optimal
locally recoverable codes and small dimension optimal (r, δ) locally
recoverable codes with any fixed given minimum distance.
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2208.01138v2
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