In this paper, we give many new Singleton type upper bounds on the sizes of codes with given minimum Hamming distances. These upper bounds are stronger than the Griesmer bound when the lengths of codes are large. Some upper bounds on the lengths of general small Singleton defect codes are presented. Our generalized Singleton type upper bounds have wide applications to symbol-pair codes, insertion-deletion codes and locally recoverable codes. The generalized Singleton type upper bounds on symbol-pair codes and insertion-deletion codes are much stronger than the direct Singleton bounds on symbol-pair codes and insertion-deletion codes when the lengths are large and the Hamming minimum distances are small. Upper bounds on the lengths of small dimension optimal locally recoverable codes and small dimension optimal$(r, \delta)$locally recoverable codes with any given minimum distance are also presented.

For a linear locally recoverable (LRC) code with length n, dimension k and locality r, its minimum distance d satisfies d ≤ n-k+2-⌈k/r⌉. A code attaining this bound is called optimal. Many families of optimal locally recoverable codes have been constructed by using different techniques in finite fields or algebraic curves. However only optimal LRC codes with lengths n >> q and minimum distances restricted to few constants smaller than 9, have been given in previous constructions. No optimal LRC code over a general finite field Fqwith the length n ~ q2and the minimum distance d ≥ 9 has been constructed. In this article we present a general construction of optimal LRC codes over arbitrary finite fields. Over any given finite field Fq, for any given r ∈ {1,2,...,q-1} and given d satisfying 3 ≤ d ≤ min{r+1,q+1-r}, we construct explicitly an optimal LRC code with length n=q(r+1), locality r and minimum distance d. We also give an asymptotic bound of q-ary LRC codes with locality r (r ≤ q-1), which is better than some known previous asymptotic bounds in some parameter range. Moreover many long LRC codes with locality r (r ≤ q-1) and small defect s=n-k+2-⌈k/r⌉-d are also constructed from algebraic curves with many rational points.

A basic problem for the constant dimension subspace coding is to determine the maximal possible size Aq(n, d, k) of a set of k-dimensional subspaces in Fnqsuch that the subspace distance satisfies d(U, V ) = 2k - 2 dim (U ∩ V ) ≥ d for any two different subspaces U and V in this set. We present two new constructions of constant dimension subspace codes using subsets of maximum rank-distance (MRD) codes in several blocks. This method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting MRD codes. In these two constructions, subsets of MRD codes with bounded ranks play an essential role. The Delsarte theorem about the rank distribution of MRD codes is an important ingredient to count codewords in our constructed constant dimension subspace codes. We give many new lower bounds for Aq(n, d, k). More than 110 new constant dimension subspace codes better than previously best known codes are constructed.

The main problem of constant-dimension subspace coding is to determine the maximal possible size Aq(n, d, k) of a set of k-dimensional subspaces in Fnq such that the subspace distance satisfies d(U, V) ≥ d for any two different subspaces U and V in this set. In this paper, we give a direct construction of constant-dimension subspace codes from two parallel versions of maximum rank-distance codes. The problem about the sizes of our constructed constant-dimension subspace codes is transformed into finding a suitable sufficient condition to restrict number of the roots of L1(L2(x)) - x where L1and L2are q-polynomials over the extension field Fqn. New lower bounds for Aq(4k, 2k, 2k), Aq(4k t 2, 2k, 2k t 1), and Aq(4k t 2, 2(k - 1), 2k t 1) are presented. Many new constantdimension subspace codes better than previously best known codes with small parameters are constructed.