Optimal locally repairable codes of distance 33 and 44 via cyclic codes

Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code {\it optimal} if it achieves the Singleton-type bound). In the breakthrough work of \cite{TB14}, several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in \cite{TB14} are upper bounded by the code alphabet size $q$. Recently, it was proved through extension of construction in \cite{TB14} that length of $q$-ary optimal locally repairable codes can be $q+1$ in \cite{JMX17}. Surprisingly, \cite{BHHMV16} presented a few examples of $q$-ary optimal locally repairable codes of small distance and locality with code length achieving roughly $q^2$. Very recently, it was further shown in \cite{LMX17} that there exist $q$-ary optimal locally repairable codes with length bigger than $q+1$ and distance propositional to $n$. Thus, it becomes an interesting and challenging problem to construct new families of $q$-ary optimal locally repairable codes of length bigger than $q+1$. In this paper, we construct a class of optimal locally repairable codes of distance $3$ and $4$ with unbounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen

Cyclic LRC Codes, binary LRC codes, and upper bounds on the distance of cyclic codes

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalize the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, no. 8, 2014). In this paper we focus on optimal cyclic codes that arise from this construction. We give a characterization of these codes in terms of their zeros, and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance. The locality parameter of a cyclic code is related to the dual distance of this code, and we phrase our results in terms of upper bounds on the dual distance.Comment: 12pp., submitted for publication. An extended abstract of this submission was posted earlier as arXiv:1502.01414 and was published in Proceedings of the 2015 IEEE International Symposium on Information Theory, Hong Kong, China, June 14-19, 2015, pp. 1262--126

Optimal Linear and Cyclic Locally Repairable Codes over Small Fields

We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the first construction has availability one, the second binary code is characterized by multiple available repair sets based on a binary Simplex code. The third approach extends the first one to q-ary cyclic codes including (binary) extension fields, where the locality property is determined by the properties of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear codes with locality greater than two are obtained by the fourth construction.Comment: IEEE Information Theory Workshop (ITW) 2015, Apr 2015, Jerusalem, Israe

A Study on the Impact of Locality in the Decoding of Binary Cyclic Codes

In this paper, we study the impact of locality on the decoding of binary cyclic codes under two approaches, namely ordered statistics decoding (OSD) and trellis decoding. Given a binary cyclic code having locality or availability, we suitably modify the OSD to obtain gains in terms of the Signal-To-Noise ratio, for a given reliability and essentially the same level of decoder complexity. With regard to trellis decoding, we show that careful introduction of locality results in the creation of cyclic subcodes having lower maximum state complexity. We also present a simple upper-bounding technique on the state complexity profile, based on the zeros of the code. Finally, it is shown how the decoding speed can be significantly increased in the presence of locality, in the moderate-to-high SNR regime, by making use of a quick-look decoder that often returns the ML codeword.Comment: Extended version of a paper submitted to ISIT 201