American Institute of Mathematical Sciences

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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

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. Google Scholar

[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. Google Scholar

[3]

S. Kim, Linear algebra from module theory perspective,, available online at , ().   Google Scholar

[4]

R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, Cambridge, 1986.  Google Scholar

[5]

S. Liu and F. Oggier, On storage codes allowing partially collaborative repairs, in IEEE Int. Symp. Inf. Theory, 2014, 2440-2444. Google Scholar

[6]

S. Liu and F. Oggier, Two storage code constructions allowing partially collaborative repairs, in Int. Symp. Inf. Theory Appl., IEEE, 2014, 378-382. Google Scholar

[7]

S. Liu and F. Oggier, On the Design of Storage Orbit Codes, in Coding Theory and Applications, Springer, 2015, 263-271. Google Scholar

[8]

F. Oggier, Some constructions of storage codes from Grassmann graphs, in Proc. Int. Zurich Sem. Commun., Zürich, 2014. Google Scholar

[9]

F. Oggier and A. Datta, Self-repairing codes for distributed storage - a projective geometric construction, in IEEE Inf. Theory Workshop, 2011, 30-34. Google Scholar

[10]

F. Oggier and A. Datta, Coding Techniques for Repairability in Networked Distributed Storage Systems, Now Publishers, 2013. Google Scholar

[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. Google Scholar

[12]

N. Raviv and T. Etzion, Distributed storage systems based on equidistant subspace codes,, preprint, ().   Google Scholar

[13]

K. W. Shum and Y. Hu, Cooperative regenerating codes, IEEE Trans. Inf. Theory, {59} (2013), 7229-7258. doi: 10.1109/TIT.2013.2274265.  Google Scholar

[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.  Google Scholar

[15]

Z. Wang, I. Tamo and J. Bruck, Long MDS codes for optimal repair bandwidth, in IEEE Int. Symp. Inf. Theory, 2012, 1182-1186. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

S. Kim, Linear algebra from module theory perspective,, available online at , ().   Google Scholar

[4]

R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, Cambridge, 1986.  Google Scholar

[5]

S. Liu and F. Oggier, On storage codes allowing partially collaborative repairs, in IEEE Int. Symp. Inf. Theory, 2014, 2440-2444. Google Scholar

[6]

S. Liu and F. Oggier, Two storage code constructions allowing partially collaborative repairs, in Int. Symp. Inf. Theory Appl., IEEE, 2014, 378-382. Google Scholar

[7]

S. Liu and F. Oggier, On the Design of Storage Orbit Codes, in Coding Theory and Applications, Springer, 2015, 263-271. Google Scholar

[8]

F. Oggier, Some constructions of storage codes from Grassmann graphs, in Proc. Int. Zurich Sem. Commun., Zürich, 2014. Google Scholar

[9]

F. Oggier and A. Datta, Self-repairing codes for distributed storage - a projective geometric construction, in IEEE Inf. Theory Workshop, 2011, 30-34. Google Scholar

[10]

F. Oggier and A. Datta, Coding Techniques for Repairability in Networked Distributed Storage Systems, Now Publishers, 2013. Google Scholar

[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. Google Scholar

[12]

N. Raviv and T. Etzion, Distributed storage systems based on equidistant subspace codes,, preprint, ().   Google Scholar

[13]

K. W. Shum and Y. Hu, Cooperative regenerating codes, IEEE Trans. Inf. Theory, {59} (2013), 7229-7258. doi: 10.1109/TIT.2013.2274265.  Google Scholar

[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.  Google Scholar

[15]

Z. Wang, I. Tamo and J. Bruck, Long MDS codes for optimal repair bandwidth, in IEEE Int. Symp. Inf. Theory, 2012, 1182-1186. Google Scholar

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