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doi: 10.3934/amc.2020099
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Locally repairable codes with high availability based on generalised quadrangles

Michel Lavrauw 1, and  Geertrui Van de Voorde 2,,

1. 

Faculty of Engineering and Natural Sciences, Sabancı University, 34956 Orhanlı, Tuzla, Istanbul, Turkey
2. 

School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, 8140 Christchurch, New Zealand

* Corresponding author: Geertrui Van de Voorde

Received  January 2020 Revised  May 2020 Early access July 2020

Locally Repairable Codes (LRC's) based on generalised quadrangles were introduced by Pamies-Juarez, Hollmann and Oggier in [3], and bounds on the repairability and availability were derived. In this paper, we determine the values of the repairability and availability of such LRC's for a large portion of the currently known generalised quadrangles. In order to do so, we determine the minimum weight of the codes of translation generalised quadrangles and characterise the codewords of minimum weight.

Keywords: linear code, LRC code, generalised quadrangle, egg.
Mathematics Subject Classification: 51E12, 51E20, 94B05.
Citation: Michel Lavrauw, Geertrui Van de Voorde. Locally repairable codes with high availability based on generalised quadrangles. Advances in Mathematics of Communications, doi: 10.3934/amc.2020099
References:
[1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.  Google Scholar
[2]

B. Bagchi and N. S. N. Sastry, Codes associated with generalized polygons, Geom. Dedicata, 27 (1988), 1-8.  doi: 10.1007/BF00181609.  Google Scholar

[3]

L. Pamies-Juarez, H. D. L. Hollmann and F. Oggier, Locally repairable codes with multiple repair alternatives, In IEEE International Symposium on Information Theory - Proceedings (ISIT), 2013,892–896. doi: 10.1109/ISIT.2013.6620355.  Google Scholar

[4]

P. Delsarte, J. M. Goethals and F. J. MacWilliams, On generalized Reed-Muller codes and their relatives, Information and Control, 16 (1970), 403-442.  doi: 10.1016/S0019-9958(70)90214-7.  Google Scholar

[5]

P. Gopalan, C. Huang, H. Simitci and and S. Yekhanin, On the locality of codeword symbols, IEEE Trans. Inform. Theory, 58 (2012), 6925-6934.  doi: 10.1109/TIT.2012.2208937.  Google Scholar

[6]

S. J. Johnson and S. R. Weller, Codes for iterative decoding from partial geometries, IEEE Trans. Comm., 52 (2004), 236-243.  doi: 10.1109/ISIT.2002.1023582.  Google Scholar

[7]

T. Kasami and N. Tokura, On the weight structure of Reed-Muller codes, IEEE Trans. Inf. Theory, 16 (1970), 752-759.  doi: 10.1109/tit.1970.1054545.  Google Scholar

[8]

J. L. Kim, K. E. Mellinger and L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr., 42 (2007), 73-92.  doi: 10.1007/s10623-006-9017-6.  Google Scholar

[9]

M. Lavrauw, Scattered Subspaces with Respect to Spreads and Eggs in Finite Projective Spaces, PhD thesis, Technical University of Eindhoven, The Netherlands, 2001.  Google Scholar

[10]

M. Lavrauw and G. Van de Voorde, Field reduction and linear sets in finite geometry, Topics in Finite Fields, Contemp. Math., volume 632, Amer. Math. Soc., Providence, RI, 2015,271–293. doi: 10.1090/conm/632/12633.  Google Scholar

[11]

S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman Advanced Publishing Program, Boston, MA, 1984.  Google Scholar

[12]

V. Pepe, L. Storme and G. Van de Voorde, Small weight codewords in the LDPC codes arising from linear representations of geometries, J. Combin. Des., 17 (2009), 1-24.  doi: 10.1002/jcd.20179.  Google Scholar

[13]

R. Rolland, The second weight of generalized Reed–Muller codes in most cases, Cryptogr. Commun., 2 (2010), 19-40.  doi: 10.1007/s12095-009-0014-2.  Google Scholar

[14]

I. Tamo, A. Barg and A. Frolov, Bounds on the parameters of locally recoverable codes, IEEE Trans. Inform. Theory, 62 (2016), 3070-3083.  doi: 10.1109/TIT.2016.2518663.  Google Scholar

[15]

G. Van de Voorde, Blocking Sets in Finite Projective Spaces and Coding Theory, PhD thesis, Ghent University, Belgium, 2010. Google Scholar

show all references

References:
[1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.  Google Scholar
[2]

B. Bagchi and N. S. N. Sastry, Codes associated with generalized polygons, Geom. Dedicata, 27 (1988), 1-8.  doi: 10.1007/BF00181609.  Google Scholar

[3]

L. Pamies-Juarez, H. D. L. Hollmann and F. Oggier, Locally repairable codes with multiple repair alternatives, In IEEE International Symposium on Information Theory - Proceedings (ISIT), 2013,892–896. doi: 10.1109/ISIT.2013.6620355.  Google Scholar

[4]

P. Delsarte, J. M. Goethals and F. J. MacWilliams, On generalized Reed-Muller codes and their relatives, Information and Control, 16 (1970), 403-442.  doi: 10.1016/S0019-9958(70)90214-7.  Google Scholar

[5]

P. Gopalan, C. Huang, H. Simitci and and S. Yekhanin, On the locality of codeword symbols, IEEE Trans. Inform. Theory, 58 (2012), 6925-6934.  doi: 10.1109/TIT.2012.2208937.  Google Scholar

[6]

S. J. Johnson and S. R. Weller, Codes for iterative decoding from partial geometries, IEEE Trans. Comm., 52 (2004), 236-243.  doi: 10.1109/ISIT.2002.1023582.  Google Scholar

[7]

T. Kasami and N. Tokura, On the weight structure of Reed-Muller codes, IEEE Trans. Inf. Theory, 16 (1970), 752-759.  doi: 10.1109/tit.1970.1054545.  Google Scholar

[8]

J. L. Kim, K. E. Mellinger and L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr., 42 (2007), 73-92.  doi: 10.1007/s10623-006-9017-6.  Google Scholar

[9]

M. Lavrauw, Scattered Subspaces with Respect to Spreads and Eggs in Finite Projective Spaces, PhD thesis, Technical University of Eindhoven, The Netherlands, 2001.  Google Scholar

[10]

M. Lavrauw and G. Van de Voorde, Field reduction and linear sets in finite geometry, Topics in Finite Fields, Contemp. Math., volume 632, Amer. Math. Soc., Providence, RI, 2015,271–293. doi: 10.1090/conm/632/12633.  Google Scholar

[11]

S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman Advanced Publishing Program, Boston, MA, 1984.  Google Scholar

[12]

V. Pepe, L. Storme and G. Van de Voorde, Small weight codewords in the LDPC codes arising from linear representations of geometries, J. Combin. Des., 17 (2009), 1-24.  doi: 10.1002/jcd.20179.  Google Scholar

[13]

R. Rolland, The second weight of generalized Reed–Muller codes in most cases, Cryptogr. Commun., 2 (2010), 19-40.  doi: 10.1007/s12095-009-0014-2.  Google Scholar

[14]

I. Tamo, A. Barg and A. Frolov, Bounds on the parameters of locally recoverable codes, IEEE Trans. Inform. Theory, 62 (2016), 3070-3083.  doi: 10.1109/TIT.2016.2518663.  Google Scholar

[15]

G. Van de Voorde, Blocking Sets in Finite Projective Spaces and Coding Theory, PhD thesis, Ghent University, Belgium, 2010. Google Scholar

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