Generic Decoding in the Sum-Rank Metric

We propose the first non-trivial generic decoding algorithm for codes in the sum-rank metric. The new method combines ideas of well-known generic decoders in the Hamming and rank metric. For the same code parameters and number of errors, the new generic decoder has a larger expected complexity than...

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Published inIEEE transactions on information theory Vol. 68; no. 8; pp. 5075 - 5097
Main Authors Puchinger, Sven, Renner, Julian, Rosenkilde, Johan
Format Journal Article
LanguageEnglish
Published New York IEEE 01.08.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects
Algorithms
Codes
Complexity theory
Decisional sum-rank syndrome decoding problem
Decoders
Decoding
erasure decoding
Europe
generic decoding
Linear codes
Measurement
probabilistic hardness reduction
sum-rank-metric codes
Technological innovation
Upper bounds
Online AccessGet full text
ISSN0018-9448
1557-9654
DOI10.1109/TIT.2022.3167629

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Abstract We propose the first non-trivial generic decoding algorithm for codes in the sum-rank metric. The new method combines ideas of well-known generic decoders in the Hamming and rank metric. For the same code parameters and number of errors, the new generic decoder has a larger expected complexity than the known generic decoders for the Hamming metric and smaller than the known rank-metric decoders. Furthermore, we give a formal hardness reduction, providing evidence that generic sum-rank decoding is computationally hard. As a by-product of the above, we solve some fundamental coding problems in the sum-rank metric: we give an algorithm to compute the exact size of a sphere of a given sum-rank radius, and also give an upper bound as a closed formula; and we study erasure decoding with respect to two different notions of support.
AbstractList We propose the first non-trivial generic decoding algorithm for codes in the sum-rank metric. The new method combines ideas of well-known generic decoders in the Hamming and rank metric. For the same code parameters and number of errors, the new generic decoder has a larger expected complexity than the known generic decoders for the Hamming metric and smaller than the known rank-metric decoders. Furthermore, we give a formal hardness reduction, providing evidence that generic sum-rank decoding is computationally hard. As a by-product of the above, we solve some fundamental coding problems in the sum-rank metric: we give an algorithm to compute the exact size of a sphere of a given sum-rank radius, and also give an upper bound as a closed formula; and we study erasure decoding with respect to two different notions of support.
Author Rosenkilde, Johan
Puchinger, Sven
Renner, Julian
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Snippet We propose the first non-trivial generic decoding algorithm for codes in the sum-rank metric. The new method combines ideas of well-known generic decoders in...
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SubjectTerms Algorithms
Codes
Complexity theory
Decisional sum-rank syndrome decoding problem
Decoders
Decoding
erasure decoding
Europe
generic decoding
Linear codes
Measurement
probabilistic hardness reduction
sum-rank-metric codes
Technological innovation
Upper bounds
Title Generic Decoding in the Sum-Rank Metric
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