Theoretical Overview on the Flavor Issues of Massive Neutrinos

We present an overview on some basic properties of massive neutrinos and focus on their flavor issues, including the mass spectrum, flavor mixing pattern and CP violation. The lepton flavor structures are explored by taking account of the observed value of the smallest neutrino mixing angle \theta_{13}. The impact of \theta_{13} on the running behaviors of other flavor mixing parameters is discussed in some detail. The seesaw-induced enhancement of the electromagnetic dipole moments for three Majorana neutrinos is also discussed in a TeV seesaw scenario.Comment: References added. arXiv admin note: text overlap with arXiv:1203.1672, arXiv:1201.2543, arXiv:1203.311

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