Constructions and Weight Distributions of Optimal Locally Repairable Codes

Locally repairable codes (LRCs) are important for distributed storage systems due to their efficient repairing ability of the failed storage nodes. A
$q$
-ary optimal
$(n,k,r)$
-LRC is an
$[n,k,d]$
linear code over
$\mathbb {F}_{q}$
such that every code symbol has locality
$r$
, and the minimum distance attains the well-known Singleton-like bound. In this paper, we study the maximal code length, code constructions and weight distributions of
$q$
-ary optimal LRCs with locality 2 and distance 5, which are of both practical and theoretical interest. Firstly, it is proved that when the code dimension is even or odd, corresponding maximal code lengths of such
$q$
-ary optimal LRCs are
$3 \cdot \lfloor \frac {q+1}{3} \rfloor $
and
$3 \cdot \left \lfloor{ \frac {q-1}{3} }\right \rfloor +5$
, respectively. Up to the equivalence of linear codes, we propose constructions of all the possible
$q$
-ary optimal LRCs with locality 2, distance 5 and maximal code length. Then, by characterizing the weight type hierarchy of codewords, we show that the weight distribution of any
$q$
-ary optimal LRC with locality 2, distance 5 and even code dimension can be uniquely determined and explicit expression of the weight distribution is given. Moreover, it is shown that all
$q$
-ary optimal LRCs with locality 2, distance 5 and even code dimension are maximally recoverable.