Generic Decoding in the Sum-Rank Metric

• Published in: IEEE transactions on information theory Vol. 68; no. 8; pp. 5075 - 5097
• Main Authors: Puchinger, Sven, Renner, Julian, Rosenkilde, Johan
• Format: Journal Article
• Language: English
• 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
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2022.3167629

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.