On applications of orbit codes to storage

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February 2016, 10(1): 113-130. doi: 10.3934/amc.2016.10.113

On applications of orbit codes to storage

Shiqiu Liu ^1, and Frédérique Oggier ^1,

1.	Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

Received December 2014 Revised July 2015 Published March 2016

Abstract
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We consider orbit codes, namely subspace codes obtained from orbits of the action of subgroups of $GL_n(\mathbb{F}_q)$ on $m$-dimensional subspaces of $\mathbb{F}_q^n$. We discuss their applicability to design storage codes, in particular in the context of collaborative repair: we identify known storage codes that can be interpreted as orbit codes, motivate why orbit codes provide good candidates for storage codes, and translate the storage code parameters into those of the algebraic objects involved. Two simple families of storage orbit codes are given.

Keywords: orbit codes, storage codes., partial repair, Group action, subspace codes.

Mathematics Subject Classification: Primary 20H20; Secondary 11T7.

Citation: Shiqiu Liu, Frédérique Oggier. On applications of orbit codes to storage. Advances in Mathematics of Communications, 2016, 10 (1) : 113-130. doi: 10.3934/amc.2016.10.113

References:

[1]	H. D. L. Hollmann, Storage codes - coding rate and repair locality, in Int. Conf. Comp. Netw. Commun., 2013, 830-834.
[2]	A.-M. Kermarrec, N. Le Scouarnec and G. Straub, Repairing multiple failures with coordinated and adaptive regenerating codes, in Int. Symp. Network Coding, IEEE, 2011, 1-6.
[3]	S. Kim, Linear algebra from module theory perspective, available online at http://www.math.ucla.edu/~i707107/Linear Algebra.pdf
[4]	R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, Cambridge, 1986.
[5]	S. Liu and F. Oggier, On storage codes allowing partially collaborative repairs, in IEEE Int. Symp. Inf. Theory, 2014, 2440-2444.
[6]	S. Liu and F. Oggier, Two storage code constructions allowing partially collaborative repairs, in Int. Symp. Inf. Theory Appl., IEEE, 2014, 378-382.
[7]	S. Liu and F. Oggier, On the Design of Storage Orbit Codes, in Coding Theory and Applications, Springer, 2015, 263-271.
[8]	F. Oggier, Some constructions of storage codes from Grassmann graphs, in Proc. Int. Zurich Sem. Commun., Zürich, 2014.
[9]	F. Oggier and A. Datta, Self-repairing codes for distributed storage - a projective geometric construction, in IEEE Inf. Theory Workshop, 2011, 30-34.
[10]	F. Oggier and A. Datta, Coding Techniques for Repairability in Networked Distributed Storage Systems, Now Publishers, 2013.
[11]	K. V. Rashmi, N. B. Shah, P. V. Kumar and K. Ramchandran, Explicit construction of optimal exact regenerating codes for distributed storage, in 47th Ann. Allerton Conf. Commun. Control Comput., IEEE, 2009, 1243-1249.
[12]	N. Raviv and T. Etzion, Distributed storage systems based on equidistant subspace codes, preprint, arXiv:1406.6170
[13]	K. W. Shum and Y. Hu, Cooperative regenerating codes, IEEE Trans. Inf. Theory, {59} (2013), 7229-7258. doi: 10.1109/TIT.2013.2274265.
[14]	A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inf. Theory, 59 (2013), 7386-7404. doi: 10.1109/TIT.2013.2274266.
[15]	Z. Wang, I. Tamo and J. Bruck, Long MDS codes for optimal repair bandwidth, in IEEE Int. Symp. Inf. Theory, 2012, 1182-1186.

show all references

References:

[1]	H. D. L. Hollmann, Storage codes - coding rate and repair locality, in Int. Conf. Comp. Netw. Commun., 2013, 830-834.
[2]	A.-M. Kermarrec, N. Le Scouarnec and G. Straub, Repairing multiple failures with coordinated and adaptive regenerating codes, in Int. Symp. Network Coding, IEEE, 2011, 1-6.
[3]	S. Kim, Linear algebra from module theory perspective, available online at http://www.math.ucla.edu/~i707107/Linear Algebra.pdf
[4]	R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, Cambridge, 1986.
[5]	S. Liu and F. Oggier, On storage codes allowing partially collaborative repairs, in IEEE Int. Symp. Inf. Theory, 2014, 2440-2444.
[6]	S. Liu and F. Oggier, Two storage code constructions allowing partially collaborative repairs, in Int. Symp. Inf. Theory Appl., IEEE, 2014, 378-382.
[7]	S. Liu and F. Oggier, On the Design of Storage Orbit Codes, in Coding Theory and Applications, Springer, 2015, 263-271.
[8]	F. Oggier, Some constructions of storage codes from Grassmann graphs, in Proc. Int. Zurich Sem. Commun., Zürich, 2014.
[9]	F. Oggier and A. Datta, Self-repairing codes for distributed storage - a projective geometric construction, in IEEE Inf. Theory Workshop, 2011, 30-34.
[10]	F. Oggier and A. Datta, Coding Techniques for Repairability in Networked Distributed Storage Systems, Now Publishers, 2013.
[11]	K. V. Rashmi, N. B. Shah, P. V. Kumar and K. Ramchandran, Explicit construction of optimal exact regenerating codes for distributed storage, in 47th Ann. Allerton Conf. Commun. Control Comput., IEEE, 2009, 1243-1249.
[12]	N. Raviv and T. Etzion, Distributed storage systems based on equidistant subspace codes, preprint, arXiv:1406.6170
[13]	K. W. Shum and Y. Hu, Cooperative regenerating codes, IEEE Trans. Inf. Theory, {59} (2013), 7229-7258. doi: 10.1109/TIT.2013.2274265.
[14]	A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inf. Theory, 59 (2013), 7386-7404. doi: 10.1109/TIT.2013.2274266.
[15]	Z. Wang, I. Tamo and J. Bruck, Long MDS codes for optimal repair bandwidth, in IEEE Int. Symp. Inf. Theory, 2012, 1182-1186.

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