Document

# List Decoding of Rank-Metric Codes with Row-To-Column Ratio Bigger Than 1/2

## File

LIPIcs.ICALP.2023.89.pdf
• Filesize: 0.93 MB
• 14 pages

## Cite As

Shu Liu, Chaoping Xing, and Chen Yuan. List Decoding of Rank-Metric Codes with Row-To-Column Ratio Bigger Than 1/2. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 89:1-89:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.89

## Abstract

Despite numerous results about the list decoding of Hamming-metric codes, development of list decoding on rank-metric codes is not as rapid as its counterpart. The bound of list decoding obeys the Gilbert-Varshamov bound in both the metrics. In the case of the Hamming-metric, the Gilbert-Varshamov bound is a trade-off among rate, decoding radius and alphabet size, while in the case of the rank-metric, the Gilbert-Varshamov bound is a trade-off among rate, decoding radius and column-to-row ratio (i.e., the ratio between the numbers of columns and rows). Hence, alphabet size and column-to-row ratio play a similar role for list decodability in each metric. In the case of the Hamming-metric, it is more challenging to list decode codes over smaller alphabets. In contrast, in the case of the rank-metric, it is more difficult to list decode codes with large column-to-row ratio. In particular, it is extremely difficult to list decode square matrix rank-metric codes (i.e., the column-to-row ratio is equal to 1). The main purpose of this paper is to explicitly construct a class of rank-metric codes 𝒞 of rate R with the column-to-row ratio up to 2/3 and efficiently list decode these codes with decoding radius beyond the decoding radius (1-R)/2 (note that (1-R)/2 is at least half of relative minimum distance δ). In literature, the largest column-to-row ratio of rank-metric codes that can be efficiently list decoded beyond half of minimum distance is 1/2. Thus, it is greatly desired to efficiently design list decoding algorithms for rank-metric codes with the column-to-row ratio bigger than 1/2 or even close to 1. Our key idea is to compress an element of the field F_qⁿ into a smaller F_q-subspace via a linearized polynomial. Thus, the column-to-row ratio gets increased at the price of reducing the code rate. Our result shows that the compression technique is powerful and it has not been employed in the topic of list decoding of both the Hamming and rank metrics. Apart from the above algebraic technique, we follow some standard techniques to prune down the list. The algebraic idea enables us to pin down the message into a structured subspace of dimension linear in the number n of columns. This "periodic" structure allows us to pre-encode the message to prune down the list.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Coding theory
##### Keywords
• Coding theory
• List-decoding
• Rank-metric codes

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Philippe Delsarte. Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory, Ser. A, 25(3):226-241, 1978. URL: https://doi.org/10.1016/0097-3165(78)90015-8.
2. Yang Ding. On list-decodability of random rank metric codes and subspace codes. IEEE Trans. Inf. Theory, 61(1):51-59, 2015. URL: https://doi.org/10.1109/TIT.2014.2371915.
3. Michael A. Forbes and Amir Shpilka. On identity testing of tensors, low-rank recovery and compressed sensing. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 163-172. ACM, 2012. URL: https://doi.org/10.1145/2213977.2213995.
4. Ernst Gabidulin. Theory of codes with maximum rank distance (translation). Problems of Information Transmission, 21:1-12, January 1985.
5. Venkatesan Guruswami. List decoding of error correcting codes. PhD thesis, Massachusetts Institute of Technology, 2001. URL: http://dspace.mit.edu/handle/1721.1/8700.
6. Venkatesan Guruswami and Swastik Kopparty. Explicit subspace designs. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 608-617. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.71.
7. Venkatesan Guruswami and Atri Rudra. Better binary list decodable codes via multilevel concatenation. IEEE Trans. Inf. Theory, 55(1):19-26, 2009. URL: https://doi.org/10.1109/TIT.2008.2008124.
8. Venkatesan Guruswami and Carol Wang. Linear-algebraic list decoding for variants of reed-solomon codes. IEEE Trans. Inf. Theory, 59(6):3257-3268, 2013. URL: https://doi.org/10.1109/TIT.2013.2246813.
9. Venkatesan Guruswami, Carol Wang, and Chaoping Xing. Explicit list-decodable rank-metric and subspace codes via subspace designs. IEEE Trans. Inf. Theory, 62(5):2707-2718, 2016. URL: https://doi.org/10.1109/TIT.2016.2544347.
10. Venkatesan Guruswami and Chaoping Xing. Folded codes from function field towers and improved optimal rate list decoding. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 339-350. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214009.
11. Venkatesan Guruswami and Chaoping Xing. List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 843-852. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488715.
12. Venkatesan Guruswami and Chaoping Xing. Optimal rate list decoding over bounded alphabets using algebraic-geometric codes. J. ACM, 69(2):10:1-10:48, 2022. URL: https://doi.org/10.1145/3506668.
13. Venkatesan Guruswami, Chaoping Xing, and Chen Yuan. Subspace designs based on algebraic function fields. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 86:1-86:10. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.86.
14. Ralf Koetter and Frank R. Kschischang. Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory, 54(8):3579-3591, 2008. URL: https://doi.org/10.1109/TIT.2008.926449.
15. Antonia Wachter-Zeh. Bounds on list decoding of rank-metric codes. IEEE Trans. Inf. Theory, 59(11):7268-7277, 2013. URL: https://doi.org/10.1109/TIT.2013.2274653.
16. Huaxiong Wang, Chaoping Xing, and Reihaneh Safavi-Naini. Linear authentication codes: bounds and constructions. IEEE Trans. Inf. Theory, 49(4):866-872, 2003. URL: https://doi.org/10.1109/TIT.2003.809567.
17. Chaoping Xing and Chen Yuan. A new class of rank-metric codes and their list decoding beyond the unique decoding radius. IEEE Trans. Inf. Theory, 64(5):3394-3402, 2018. URL: https://doi.org/10.1109/TIT.2017.2780848.