Repeated-root Constacyclic Codes with Optimal Locality

A code is called a locally repairable code (LRC) if any code symbol is a function of a small fraction of other code symbols. When a locally repairable code is employed in a distributed storage systems, an erased symbol can be recovered by accessing only a small number of other symbols, and hence alleviating the network resources required during the repair process. In this paper we consider repeatedroot constacyclic codes, which is a generalization of cyclic codes, that are optimal with respect to a Singleton-like bound on minimum distance. An LRC with the structure of a constacyclic code can be encoded efficiently using any encoding algorithm for constacyclic codes in general. In this paper we obtain optimal LRCs among these repeated-root constacyclic codes. Several infinite classes of optimal LRCs over a fixed alphabet are found. Under a further assumption that the ambient space of the repeated-root constacyclic codes is a chain ring, we show that there is no other optimal LRC.

[1]  Hai Q. Dinh,et al.  Repeated-root constacyclic codes of length 2ps , 2012, Finite Fields Their Appl..

[2]  Yuan Luo,et al.  Optimal Locally Repairable Codes of Distance 3 and 4 via Cyclic Codes , 2019, IEEE Transactions on Information Theory.

[3]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[4]  Sriram Vishwanath,et al.  Optimal locally repairable codes via rank-metric codes , 2013, 2013 IEEE International Symposium on Information Theory.

[5]  Gretchen L. Matthews,et al.  Locally recoverable codes from algebraic curves and surfaces , 2017, ArXiv.

[6]  Ferruh Özbudak,et al.  A note on negacyclic and cyclic codes of length ps over a finite field of characteristic p , 2009, Adv. Math. Commun..

[7]  Jianfa Qian,et al.  New Optimal Cyclic Locally Recoverable Codes of Length $n=2(q+1)$ , 2020, IEEE Transactions on Information Theory.

[8]  Shu-Tao Xia,et al.  Bounds and Constructions of Locally Repairable Codes: Parity-Check Matrix Approach , 2016, IEEE Transactions on Information Theory.

[9]  Dongdai Lin,et al.  Bounds for Binary Linear Locally Repairable Codes via a Sphere-Packing Approach , 2019, IEEE Transactions on Information Theory.

[10]  Anuradha Sharma,et al.  On the Structure and Distances of Repeated-Root Constacyclic Codes of Prime Power Lengths Over Finite Commutative Chain Rings , 2019, IEEE Transactions on Information Theory.

[11]  Venkatesan Guruswami,et al.  Explicit optimal-length locally repairable codes of distance 5 , 2018, 2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[12]  Shu-Tao Xia,et al.  Constructions of Optimal $(r,\delta)$ Locally Repairable Codes via Constacyclic Codes , 2019, IEEE Transactions on Communications.

[13]  Alexandros G. Dimakis,et al.  Network Coding for Distributed Storage Systems , 2007, IEEE INFOCOM 2007 - 26th IEEE International Conference on Computer Communications.

[14]  Venkatesan Guruswami,et al.  How Long Can Optimal Locally Repairable Codes Be? , 2019, IEEE Transactions on Information Theory.

[15]  Kannan Ramchandran,et al.  A Solution to the Network Challenges of Data Recovery in Erasure-coded Distributed Storage Systems: A Study on the Facebook Warehouse Cluster , 2013, HotStorage.

[16]  Chaoping Xing,et al.  Construction of Optimal Locally Repairable Codes via Automorphism Groups of Rational Function Fields , 2020, IEEE Transactions on Information Theory.

[17]  Itzhak Tamo,et al.  A Family of Optimal Locally Recoverable Codes , 2013, IEEE Transactions on Information Theory.

[18]  Shixin Zhu,et al.  Optimal Constacyclic Locally Repairable Codes , 2019, IEEE Communications Letters.

[19]  James L. Massey,et al.  On Repeated-root Cyclic Codes , 1991, IEEE Trans. Inf. Theory.

[20]  Shu-Tao Xia,et al.  Constructions of Optimal Cyclic (r, δ) Locally Repairable Codes , 2016, ArXiv.

[21]  Chaoping Xing,et al.  Optimal Locally Repairable Codes Via Elliptic Curves , 2017, IEEE Transactions on Information Theory.

[22]  A. Robert Calderbank,et al.  Cyclic LRC codes and their subfield subcodes , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[23]  Hongwei Liu,et al.  Some Repeated-Root Constacyclic Codes Over Galois Rings , 2017, IEEE Transactions on Information Theory.

[24]  Wei Zhao,et al.  Constacyclic codes of length klmpn over a finite field , 2018, Finite Fields Their Appl..

[25]  Luis Alfonso Lastras-Montaño,et al.  Reliable Memories with Subline Accesses , 2007, 2007 IEEE International Symposium on Information Theory.

[26]  Sergio R. López-Permouth,et al.  Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes , 2012, Finite Fields Their Appl..

[27]  Zhengchun Zhou,et al.  Optimal Cyclic Locally Repairable Codes via Cyclotomic Polynomials , 2019, IEEE Communications Letters.

[28]  Cheng Huang,et al.  On the Locality of Codeword Symbols , 2011, IEEE Transactions on Information Theory.

[29]  Hongwei Liu,et al.  Repeated-root constacyclic codes of length ℓp5 and their duals , 2014, Discret. Appl. Math..

[30]  Shixin Zhu,et al.  Repeated-root constacyclic codes of length 6lps , 2021, Adv. Math. Commun..

[31]  Arya Mazumdar,et al.  Bounds on the Size of Locally Recoverable Codes , 2015, IEEE Transactions on Information Theory.

[32]  Hai Quang Dinh,et al.  On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions , 2008, Finite Fields Their Appl..

[33]  Zhang ZhiFang,et al.  Constructions of optimal locally repairable codes over small fields , 2017 .