In this paper, we consider a wide family of λ-quasi-twisted (λ-QT) codes of index 2 and provide a bound on the minimum Hamming distance. Moreover, we give a sufficient condition for dual containing with respect to Hermitian inner product of these involved codes. As an application, some good stabilizer quantum codes over small finite fields F2 or F3 are obtained from the class of λ-QT codes.

Locally repairable codes (LRCs) are a type of new erasure codes designed for modern distributed storage systems (DSSs). In order to obtain ternary LRCs of distance 6, firstly, we propose constructions with disjoint repair groups and construct several families of LRCs with 1 ≤ r ≤ 6, where codes with 3 ≤ r ≤ 6 are obtained through a search algorithm. Then, we propose a new method to extend the length of codes without changing the distance. By employing the methods such as expansion and deletion, we obtain more LRCs from a known LRC. The resulting LRCs are optimal or near optimal in terms of the Cadambe-Mazumdar (C-M) bound.

In this paper, a special class of two-generator quasi-twisted (QT) codes with index 2 will be presented. We explore the algebraic structure of the class of QT codes and the form of their Hermitian dual codes. A sufficient condition for self-orthogonality with Hermitian inner product is derived. Using the class of Hermitian self-orthogonal QT codes, we construct two new binary quantum codes [[70, 42, 7]]2, [[78, 30, 10]]2. According to Theorem 6 of Ref.[2], we further can get 9 new binary quantum codes. So a total of 11 new binary quantum codes are obtained and there are 10 quantum codes that can break the quantum Gilbert-Varshamov (GV) bound.