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Register a new userImproved Power Decoding of Algebraic Geometry Codes
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Added by
Grigory Solomatov
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Power decoding is a partial decoding paradigm for arbitrary algebraic geometry codes for decoding beyond half the minimum distance, which usually returns the unique closest codeword, but in rare cases fails to return anything. The original version decodes roughly up to the Sudan radius, while an improved version decodes up to the Johnson radius, but has so far been described only for Reed--Solomon and one-point Hermitian codes. In this paper we show how the improved version can be applied to any algebraic geometry code.
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We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes, which greatly extends the class of a previous paper due to Munuera, Tenorio and Torres. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.
We consider families of codes obtained by lifting a base code $mathcal{C}$ through operations such as $k$-XOR applied to local views of codewords of $mathcal{C}$, according to a suitable $k$-uniform hypergraph. The $k$-XOR operation yields the direct sum encoding used in works of [Ta-Shma, STOC 2017] and [Dinur and Kaufman, FOCS 2017]. We give a general framework for list decoding such lifted codes, as long as the base code admits a unique decoding algorithm, and the hypergraph used for lifting satisfies certain expansion properties. We show that these properties are satisfied by the collection of length $k$ walks on an expander graph, and by hypergraphs corresponding to high-dimensional expanders. Instantiating our framework, we obtain list decoding algorithms for direct sum liftings on the above hypergraph families. Using known connections between direct sum and direct product, we also recover the recent results of Dinur et al. [SODA 2019] on list decoding for direct product liftings. Our framework relies on relaxations given by the Sum-of-Squares (SOS) SDP hierarchy for solving various constraint satisfaction problems (CSPs). We view the problem of recovering the closest codeword to a given word, as finding the optimal solution of a CSP. Constraints in the instance correspond to edges of the lifting hypergraph, and the solutions are restricted to lie in the base code $mathcal{C}$. We show that recent algorithms for (approximately) solving CSPs on certain expanding hypergraphs also yield a decoding algorithm for such lifted codes. We extend the framework to list decoding, by requiring the SOS solution to minimize a convex proxy for negative entropy. We show that this ensures a covering property for the SOS solution, and the condition and round approach used in several SOS algorithms can then be used to recover the required list of codewords.
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