In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However, most known constructions are over large fields with sizes close to the code length, which lead to the systems computationally expensive. Due to this, binary LRCs are of interest in practice.In this paper, we focus on binary linear LRCs with disjoint repair groups. We first derive an explicit bound for the dimension k of such codes, which can be served as a generalization of the bounds given in [11], [36], [37]. We also give several new constructions of binary LRCs with minimum distance d = 6 based on weakly independent sets and partial spreads, which are optimal with respect to our newly obtained bound. In particular, for locality r ∈ {2, 3} and minimum distance d = 6, we obtain the desired optimal binary linear LRCs with disjoint repair groups for almost all parameters.which is also called as Singleton-like bound since it degenerates to classical Singleton bound when r = k. Later, in [7], [19], the bound (1) is generalized to vector codes and nonlinear codes. Although it certainly holds for all LRCs, it is not tight in many cases. The tightness of the bound (1) is studied in [28], [34].We say an LRC is d-optimal if it satisfies bound (1) with equality for given n, k and r. For the case (r + 1)|n, d-optimal LRCs are constructed explicitly in [32] and [25] by using Reed-Solomon codes and Gabidulin codes respectively. However, both constructions are built over a finite field whose size is an exponential function of the code length n. In [30], for the same case (r + 1)|n, the authors construct a d-optimal code over a finite field of size sightly greater than n by using "good" polynomials. This construction can be extended to the case (r + 1) ∤ n with the minimum distance d ≥ n − k − ⌈ k r ⌉ + 1 which

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form x + γTr q n /q (x k ) is also presented, which generalizes two examples of [10].

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci. China Math., 2015, 58, pp. 2081-2094. Furthermore, we give two classes of permutation trinomial, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by

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In recent years, the rapidly increasing amounts of data created and processed through the internet resulted in distributed storage systems employing erasure coding based schemes. Aiming to balance the tradeoff between data recovery for correlated failures and efficient encoding and decoding, distributed storage systems employing maximally recoverable codes came up. Unifying a number of topologies considered both in theory and practice, Gopalan et al. [15] initiated the study of maximally recoverable codes for grid-like topologies.In this paper, we focus on the maximally recoverable codes that instantiate grid-like topologies Tm×n(1, b, 0). To characterize the property of codes for these topologies, we introduce the notion of pseudo-parity check matrix. Then, using the Combinatorial Nullstellensatz, we establish the first polynomial upper bound on the field size needed for achieving the maximal recoverability in topologies Tm×n(1, b, 0). And using hypergraph independent set approach, we further improve this general upper bound for topologies T4×n(1, 2, 0) and T3×n(1, 3, 0). By relating the problem to generalized Sidon sets in Fq, we also obtain non-trivial lower bounds on the field size for maximally recoverable codes that instantiate topologies T4×n(1, 2, 0) and T3×n(1, 3, 0).