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Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $
Locally repairable codes with high availability based on generalised quadrangles
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 |
Locally Repairable Codes (LRC's) based on generalised quadrangles were introduced by Pamies-Juarez, Hollmann and Oggier in [
References:
[1] |
E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836.![]() ![]() |
[2] |
B. Bagchi and N. S. N. Sastry,
Codes associated with generalized polygons, Geom. Dedicata, 27 (1988), 1-8.
doi: 10.1007/BF00181609. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
M. Lavrauw, Scattered Subspaces with Respect to Spreads and Eggs in Finite Projective Spaces, PhD thesis, Technical University of Eindhoven, The Netherlands, 2001. |
[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. |
[11] |
S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman Advanced Publishing Program, Boston, MA, 1984. |
[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. |
[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. |
[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. |
[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.![]() ![]() |
[2] |
B. Bagchi and N. S. N. Sastry,
Codes associated with generalized polygons, Geom. Dedicata, 27 (1988), 1-8.
doi: 10.1007/BF00181609. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
M. Lavrauw, Scattered Subspaces with Respect to Spreads and Eggs in Finite Projective Spaces, PhD thesis, Technical University of Eindhoven, The Netherlands, 2001. |
[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. |
[11] |
S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman Advanced Publishing Program, Boston, MA, 1984. |
[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. |
[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. |
[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. |
[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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