The design of codes for distributed storage systems that protects from node failures has been studied for years, and locally repairable code (LRC) is such a method that gives a solution for fast recovery of node failures. Linear complementary dual code (LCD code) is useful for preventing malicious attacks, which helps to secure the system. In this paper, we combine LRC and LCD code by integration of enhancing security and repair efficiency, and propose some techniques for constructing LCD codes with their localities determined. On the basis of these methods and inheriting previous achievements of optimal LCD codes, we give optimal or near-optimal [n, k, d;r] LCD codes for k≤6 and n≥k+1 with relatively small locality, mostly r≤3. Since all of our obtained codes are distance-optimal, in addition, we show that the majority of them are r-optimal and the other 63 codes are all near r-optimal, according to CM bound.

Locally repairable code (LRC) can recover any codeword symbol failure by accessing a small number of other symbols, which can increase the efficiency during the repair process. In a distributed storage system with locally repairable codes, any node failure can be rebuilt by accessing other fixed nodes. It is a promising prospect for the application of LRC. In this paper, some methods of constructing matrices which can generate codes with small locality will be proposed firstly. By analyzing the parameters, we construct the generator matrices of the best-known ternary linear codes of dimension 6, using methods such as shortening, puncturing and expansion. After analyzing the linear dependence of the column vectors in the generator matrices above, we find out the locality of the codes they generate. Many codes with small locality have been found.

In order to guarantee pairwise independence of codewords in an ensemble of convolutional codes it is necessary to consider time-varying codes. However, Seguin has shown that the pairwise independence property is not strictly necessary when applying the random coding argument and on this basis he derives a new random coding bound for rate 1/n fixed convolutional codes. In this paper we show that a similar random coding bound can be obtained for rate k/n fixed convolutional codes.