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Abstract In this paper, we study deep holes of Gabidulin codes in both rank and Hamming metrics. Specifically, first, we give a tight lower bound for the distance of any word to a Gabidulin code and a sufficient and necessary condition for achieving this lower bound as well. Then, a class of deep holes of a Gabidulin code are discovered. Furthermore, we obtain some other deep holes for certain Gabidulin codes. [ABSTRACT FROM AUTHOR]

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair costs by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefits of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code lengths. Specifically, we first obtain two classes of q -ary cyclic Singleton-optimal (n , k , d = 6 ; r = 2) -LRCs with length n = 3 (q + 1) when 3 | (q − 1) and q is even, and length n = 3 2 (q + 1) when 3 | (q − 1) and q ≡ 1 (mod 4) , respectively. To the best of our knowledge, this is the first construction of q -ary cyclic Singleton-optimal LRCs with length n > q + 1 and minimum distance d ≥ 5. On the other hand, an LRC achieving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of q -ary perfect LRCs with length n = q m − 1 q − 1 , minimum distance d = 5 and locality r = 2. [ABSTRACT FROM AUTHOR]

Let p be a prime and let q be a power of p . In this paper, by using generalized Reed–Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance-separable (MDS) codes with parameters [ [ t q , t q − 2 d + 2 , d ] ] q for any 1 ≤ t ≤ q , 2 ≤ d ≤ ⌊ t q + q − 1 q + 1 ⌋ + 1 , and [ [ t ( q + 1 ) + 2 , t ( q + 1 ) − 2 d + 4 , d ] ] q for any 1 ≤ t ≤ q − 1 , 2 ≤ d ≤ t + 2 with ( p , t , d ) ≠ ( 2 , q − 1 , q ) . Our quantum MDS codes have flexible parameters, and have minimum distances larger than q 2 + 1 when t > q 2 . Furthermore, it turns out that our constructions generalize and improve some previous results. [ABSTRACT FROM AUTHOR]

As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. This paper is devoted to the study of the q -polynomial approach to constacyclic codes. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper. [ABSTRACT FROM AUTHOR]

Locally repairable codes with locality r (r -LRCs for short) were introduced by Gopalan et al. [1] to recover a failed node of the code from at most other r available nodes. And then (r , δ) -locally repairable codes ((r , δ) -LRCs for short) were produced by Prakash et al. [2] for tolerating multiple failed nodes. An r -LRC can be viewed as an (r , 2) -LRC. An (r , δ) -LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct q -ary optimal (r , δ) -LRCs with length much larger than q. Surprisingly, Luo et al. [3] presented a construction of q -ary optimal r -LRCs of minimum distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of q) via cyclic codes. In this paper, inspired by the work of [3] , we firstly construct two classes of optimal cyclic (r , δ) -LRCs with unbounded lengths and minimum distances δ + 1 or δ + 2 , which generalize the results about the δ = 2 case given in [3]. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic (r , δ) -LRCs with unbounded length and larger minimum distance 2 δ. Furthermore, when δ = 3 , we give another class of optimal cyclic (r , 3) -LRCs with unbounded length and minimum distance 6. [ABSTRACT FROM AUTHOR]