A polymatroid approach to generalized weights of rank metric codes. (English) Zbl 1480.94048
In this work, the authors introduce and study properties of the generalized weights of a \((q,m)\)-demi-polymatroid, which is a more general concept than a \((q,m)\)-polymatroid. The authors define the generalized weights of a flag of Delsarte codes by associating a \((q,m)\)-demi-polymatroid to the flag. This generalizes a definition given by U. Martínez-Peñas and R. Matsumoto [IEEE Trans. Inf. Theory 64, No. 4, Part 1, 2529–2549 (2018; Zbl 1390.94906)] for the generalized weights of flags of length 2. The authors prove that the generalized weights of rank-metric codes introduced by A. Ravagnani [J. Pure Appl. Algebra 220, No. 5, 1946–1962 (2016; Zbl 1345.94102)] are a particular case of the generalized weights of a \((q,m)\)-demi-polymatroid.
Reviewer: Hiram López (Cleveland)
MSC:
| 94B60 | Other types of codes |
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 15A03 | Vector spaces, linear dependence, rank, lineability |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
References:
| [1] | Berhuy, G., Fasel, J., Garotta, O.: Rank weights for arbitrary finite field extensions. Adv. Math. Commun. (2020). 10.3934/amc.2020083 · Zbl 1481.94159 |
| [2] | Britz, T., Higher support matroids, Discret. Math., 307, 2300-2308 (2007) · Zbl 1121.05024 · doi:10.1016/j.disc.2006.12.001 |
| [3] | Britz, T.; Johnsen, T.; Mayhew, D.; Shiromoto, K., Wei-type duality theorems for matroids, Des. Codes Cryptogr., 62, 331-341 (2012) · Zbl 1236.05054 · doi:10.1007/s10623-011-9524-y |
| [4] | Britz, T.; Johnsen, T.; Martin, J., Chains, demi-matroids and profiles, IEEE Trans. Inform. Theory, 60, 986-991 (2014) · Zbl 1364.94591 · doi:10.1109/TIT.2013.2292524 |
| [5] | Britz, T.; Mammoliti, A.; Shiromoto, K., Wei-type duality theorems for rank metric codes, Des. Codes Cryptogr., 88, 1503-1519 (2020) · Zbl 1512.94122 · doi:10.1007/s10623-019-00688-9 |
| [6] | Delsarte, P., Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25, 226-241 (1978) · Zbl 0397.94012 · doi:10.1016/0097-3165(78)90015-8 |
| [7] | Ducoat, J.; Kyureghyan, GM; Pott, A., Generalized rank weights: a duality statement, Topics in Finite Fields, 114-123 (2015), Providence, RI: American Mathematical Society, Providence, RI |
| [8] | Gabidulin, EM, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21, 1-12 (1985) · Zbl 0585.94013 |
| [9] | Gorla, E.: Rank-Metric Codes. arXiv:1902.02650 [cs.IT] , 26 pp (2019). |
| [10] | Gorla, E.; Jurrius, R.; Lopez, HH; Ravagnani, A., Rank-metric codes and q-polymatroids, J. Algebr. Combin., 52, 1-19 (2020) · Zbl 1478.94145 · doi:10.1007/s10801-019-00889-4 |
| [11] | Jurrius, R.; Pellikaan, R., On defining generalized rank weights, Adv. Math. Commun., 11, 225-235 (2017) · Zbl 1357.94082 · doi:10.3934/amc.2017014 |
| [12] | Jurrius, R.; Pellikaan, R., Defining the \(q\)-analogue of a matroid, Electron. J. Combin., 25, 32 (2018) · Zbl 1393.05071 · doi:10.37236/6569 |
| [13] | Kurihara, J.; Matsumoto, R.; Uyematsu, T., Relative generalized rank weight of linear codes and its applications to network coding, IEEE Trans. Inform. Theory, 61, 3912-3936 (2015) · Zbl 1359.94713 · doi:10.1109/TIT.2015.2429713 |
| [14] | Larsen, A.H.: Matroider og lineære koder, Master’s thesis, Univ. Bergen, Norway, (2005). http://bora.uib.no/bitstream/handle/1956/10780/Ann-Hege-totaloppgave.pdf?sequence=1 |
| [15] | Martínez-Peñas, U.; Matsumoto, R., Relative generalized matrix weights of matrix codes for universal security on wire-tap network, IEEE Trans. Inform. Theory, 64, 2529-2548 (2018) · Zbl 1390.94906 · doi:10.1109/TIT.2017.2766292 |
| [16] | Oggier, F., Sboui, A.: On the existence of generalized rank weights. In: Proc. Int. Symp. Inf. Theory Appl., pp. 406-410, (2012). |
| [17] | Ravagnani, A., Rank-metric codes and their duality theory, Des. Codes Cryptogr., 80, 197-216 (2016) · Zbl 1348.94098 · doi:10.1007/s10623-015-0077-3 |
| [18] | Ravagnani, A., Generalized weights: an anticode approach, J. Pure Appl. Algeb., 220, 1946-1962 (2016) · Zbl 1345.94102 · doi:10.1016/j.jpaa.2015.10.009 |
| [19] | Shiromoto, K., Codes with the rank metric and matroids, Des. Codes Cryptogr., 87, 1765-1776 (2019) · Zbl 1414.05071 · doi:10.1007/s10623-018-0576-0 |
| [20] | Wei, VK, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37, 1412-1418 (1991) · Zbl 0735.94008 · doi:10.1109/18.133259 |
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