# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022010
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## On the generalised rank weights of quasi-cyclic codes

 Division of Mathematical Sciences, Nanyang Technological University, Singapore

* Corresponding author

Received  October 2021 Revised  January 2022 Early access March 2022

Generalised rank weights were formulated in analogy to Wei's generalised Hamming weights, but for the rank metric. In this paper we study the generalised rank weights of quasi-cyclic codes, a special class of linear codes usually studied for their properties in error correction over the Hamming metric. By using the algebraic structure of quasi-cyclic codes, a new upper bound on the generalised rank weights of quasi-cyclic codes is formulated, which is tighter than the known Singleton bound. Additionally, it is shown that the first generalised rank weight of self-dual $1$-generator quasi-cyclic codes is almost completely determined by the choice of ${\mathbb F}_{q^{m}}$.

Citation: Enhui Lim, Frédérique Oggier. On the generalised rank weights of quasi-cyclic codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022010
##### References:
 [1] E. F. Assmus, H. F. Mattson and R. J. Turyn, Research to Develop the Algebraic Theory of Codes, AFCRL-6700365, Air Force Cambridge Research Laboratories, Bedford, Massachusetts, 1967. [2] G. Berhuy, J. Fasel and O. Garotta, Rank weights for arbitrary finite field extensions, Advances in Mathematics of Communications, 15 (2021), 575-587.  doi: 10.3934/amc.2020083. [3] J. Ducoat, Generalized rank weights: A duality statement, Amer. Math. Soc, 632 (2015), 101-109.  doi: 10.1090/conm/632/12622. [4] J. Ducoat and F. Oggier, Rank weight hierarchy of some classes of cyclic codes, IEEE Information Theory Workshop (ITW 2014), (2014), 142–146. doi: 10.1109/ITW.2014.6970809. [5] E. Gabidulin, Theory of codes with maximum rank distance (translation), Problems of Information Transmission, 21 (1985), 3-16. [6] C. Güneri, S. Ling and B. Özkaya, Quasi-Cyclic Codes, Chap 7 of Concise Encyclopedia of Coding Theory, Chapman and Hall, CRC, 2021. [7] R. Jurrius and R. Pellikaan, On defining generalized rank weights, Adv. Math. Commun., 11 (2017), 225-235.  doi: 10.3934/amc.2017014. [8] J. Kurihara, R. Matsumoto and T. Uyematsu, Relative generalized rank weight of linear codes and its applications to network coding, IEEE Trans. Inform. Theory, 61 (2015), 3912-3936.  doi: 10.1109/TIT.2015.2429713. [9] K. Lally and P. Fitzpatrick, Algebraic structure of quasicyclic codes, Discrete Appl. Math., 111 (2001), 157-175.  doi: 10.1016/S0166-218X(00)00350-4. [10] S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes I. Finite fields, IEEE Trans. Inform. Theory, 47 (2001), 2751-2760.  doi: 10.1109/18.959257. [11] S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes III: Generator theory, IEEE Trans. Inform. Theory, 51 (2005), 2692-2700.  doi: 10.1109/TIT.2005.850142. [12] U. Martínez-Peñas, Generalized rank weights of reducible codes, optimal cases, and related properties, IEEE Trans. Inform. Theory, 64 (2018), 192-204.  doi: 10.1109/TIT.2017.2748148. [13] F. Oggier and A. Sboui, On the Existence of Generalized Rank Weights, International Symposium on Information Theory and Its Applications (ISITA 2012). [14] R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inform. Theory, 37 (1991), 328-336.  doi: 10.1109/18.75248. [15] H. Stichtenoth, On the dimension of subfield subcodes, IEEE Trans. Inform. Theory, 36 (1990), 90-93.  doi: 10.1109/18.50376. [16] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418.  doi: 10.1109/18.133259. [17] The Sage Development Team, Sage Mathematics Software (Version 9.2), http://www.sagemath.org, (2020).

show all references

##### References:
 [1] E. F. Assmus, H. F. Mattson and R. J. Turyn, Research to Develop the Algebraic Theory of Codes, AFCRL-6700365, Air Force Cambridge Research Laboratories, Bedford, Massachusetts, 1967. [2] G. Berhuy, J. Fasel and O. Garotta, Rank weights for arbitrary finite field extensions, Advances in Mathematics of Communications, 15 (2021), 575-587.  doi: 10.3934/amc.2020083. [3] J. Ducoat, Generalized rank weights: A duality statement, Amer. Math. Soc, 632 (2015), 101-109.  doi: 10.1090/conm/632/12622. [4] J. Ducoat and F. Oggier, Rank weight hierarchy of some classes of cyclic codes, IEEE Information Theory Workshop (ITW 2014), (2014), 142–146. doi: 10.1109/ITW.2014.6970809. [5] E. Gabidulin, Theory of codes with maximum rank distance (translation), Problems of Information Transmission, 21 (1985), 3-16. [6] C. Güneri, S. Ling and B. Özkaya, Quasi-Cyclic Codes, Chap 7 of Concise Encyclopedia of Coding Theory, Chapman and Hall, CRC, 2021. [7] R. Jurrius and R. Pellikaan, On defining generalized rank weights, Adv. Math. Commun., 11 (2017), 225-235.  doi: 10.3934/amc.2017014. [8] J. Kurihara, R. Matsumoto and T. Uyematsu, Relative generalized rank weight of linear codes and its applications to network coding, IEEE Trans. Inform. Theory, 61 (2015), 3912-3936.  doi: 10.1109/TIT.2015.2429713. [9] K. Lally and P. Fitzpatrick, Algebraic structure of quasicyclic codes, Discrete Appl. Math., 111 (2001), 157-175.  doi: 10.1016/S0166-218X(00)00350-4. [10] S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes I. Finite fields, IEEE Trans. Inform. Theory, 47 (2001), 2751-2760.  doi: 10.1109/18.959257. [11] S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes III: Generator theory, IEEE Trans. Inform. Theory, 51 (2005), 2692-2700.  doi: 10.1109/TIT.2005.850142. [12] U. Martínez-Peñas, Generalized rank weights of reducible codes, optimal cases, and related properties, IEEE Trans. Inform. Theory, 64 (2018), 192-204.  doi: 10.1109/TIT.2017.2748148. [13] F. Oggier and A. Sboui, On the Existence of Generalized Rank Weights, International Symposium on Information Theory and Its Applications (ISITA 2012). [14] R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inform. Theory, 37 (1991), 328-336.  doi: 10.1109/18.75248. [15] H. Stichtenoth, On the dimension of subfield subcodes, IEEE Trans. Inform. Theory, 36 (1990), 90-93.  doi: 10.1109/18.50376. [16] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418.  doi: 10.1109/18.133259. [17] The Sage Development Team, Sage Mathematics Software (Version 9.2), http://www.sagemath.org, (2020).
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