The hull of linear codes have promising utilization in coding theory and quantum coding theory. In this paper, we study the hull of generalized Reed-Solomon codes and extended generalized Reed-Solomon codes over finite fields with respect to the Euclidean inner product. Several infinite families of MDS codes with arbitrary dimensional hull are presented. As an application, using these MDS codes with arbitrary dimensional hull, we construct several new infinite families of entanglement-assisted quantum error-correcting codes with flexible parameters.

Linear codes have widespread applications in data storage systems. There are two major contributions in this paper. We first propose infinite families of optimal or distance-optimal linear codes over Fp constructed from projective spaces. Moreover, a necessary and sufficient condition for such linear codes to be Griesmer codes is presented. Secondly, as an application in data storage systems, we investigate the locality of the linear codes constructed. Furthermore, we show that these linear codes are alphabet-optimal locally repairable codes with locality 2.

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In modern practical data centers, storage nodes are usually organized into equally sized groups, which is called racks. The cost of cross-rack communication is much more expensive compared with the intra-rack communication cost. The codes for this system are called rack-aware regenerating codes. Similar to standard minimum storage regenerating (MSR) codes, it is a challenging task to construct minimum storage rack-aware regenerating (MSRR) codes achieving the cut-set bound. The known constructions of MSRR codes achieving the cut-set bound give codes with alphabet size q exponential in the code length n, more precisely, q = Ω(exp(n n )).The main contribution of this paper is to provide explicit construction of MSRR codes achieving the cut-set bound with the alphabet size linear in n. To achieve this goal, we first present a general framework to repair Reed-Solomon codes. It turns out that the known repairing schemes of Reed-Solomon codes can be realized under our general framework. Several techniques are used in this paper. In particular, we use the degree decent method to repair failure node. This technique allows us to get Reed-Solomon codes with the alphabet size linear in n. The other techniques include choice of good polynomials. Note that good polynomials are used for construction of locally repairable code in literature. To the best of our knowledge, it is the first time in this paper to make use of good polynomials for constructions of regenerating codes.

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