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Permutation polynomials (PPs) and their inverses have applications in cryptography, coding theory and combinatorial design theory. In this paper, we make a brief summary of the inverses of PPs of finite fields, and give the inverses of all PPs of degree ≤ 6 over finite fields $\mathbb {F}_{q}$ for all $q$ and the inverses of all PPs of degree 7 over $\mathbb {F}_{2^{n}}$. The explicit inverse of a class of fifth degree PPs is the main result, which is obtained by using Lucas’ theorem, some congruences of binomial coefficients, and a known formula for the inverses of PPs of finite fields. [ABSTRACT FROM AUTHOR]

The Hasse–Weil bound is a deep result in mathematics and has found wide applications in mathematics, theoretical computer science, information theory etc. In general, the bound is tight and cannot be improved. However, for some special families of curves the bound could be improved substantially. In this paper, we focus on the Hasse–Weil bound for the curve defined by y p − y = f ( x ) over the finite field F q , where p is the characteristic of F q . In 1993, Moreno–Moreno [7] gave an improvement to the Hasse–Weil bound for this family of curves. Recently, Kaufman and Lovett [4, FOCS2011] showed that the Hasse–Weil bound can be improved for this family of curves with f ( x ) = g ( x ) + h ( x ) , where g ( x ) is a polynomial of degree ≪ q and h ( x ) is a sparse polynomial of arbitrary degree but bounded weight degree. The other recent improvement by Rojas-Leon and Wan [9, Math. Ann. 2011] shows that an extra p can be removed for this family of curves if p is very large compared with polynomial degree of f ( x ) and log p ⁡ q . In this paper, we focus on the most interesting case for applications, namely p = 2 . We show that the Hasse–Weil bound for this special family of curves can be improved if q = 2 n with odd n ⩾ 3 which is the same case where Serre [10] improved the Hasse–Weil bound. However, our improvement is greater than Serre's and Moreno–Morenao's improvements for this special family of curves. Furthermore, our improvement works for p = 2 compared with the requirement of large p by Rojas-Leon and Wan. In addition, our improvement finds interesting applications to character sums, cryptography and coding theory. The key idea behind is that this curve has the Hasse–Witt invariant 0 and we show that the Hasse–Weil bound can be improved for any curves with the Hasse–Witt invariant 0. The main tool used in our proof involves Newton polygon and some results in algebraic geometry. [ABSTRACT FROM AUTHOR]

Partial MDS (PMDS) codes are a class of erasure-correcting array codes that combine local correction of the rows with global correction of the array. An $\boldsymbol {m}\times \boldsymbol {n}$ array code is called an $(\boldsymbol {r};\boldsymbol {s})$ PMDS code if each row belongs to an ${[}\boldsymbol {n},\boldsymbol {n}-\boldsymbol {r}, \boldsymbol {r}+\textbf {1}{]}$ MDS code and the code can correct erasure patterns consisting of $\boldsymbol {r}$ erasures in each row together with $\boldsymbol {s}$ more erasures anywhere in the array. While a recent construction by Calis and Koyluoglu generates $(\boldsymbol {r};\boldsymbol {s})$ PMDS codes for all $\boldsymbol {r}$ and $\boldsymbol {s}$ , its field size is exponentially large. In this paper, a family of PMDS codes with field size ${\mathcal{ O}}\left ({\max \{\boldsymbol {m},\boldsymbol {n}^{\boldsymbol {r}+\boldsymbol {s}}\}^{\boldsymbol {s}} }\right)$ is presented for the case where $\boldsymbol {r}= {\mathcal{ O}}(1), \boldsymbol {s}= {\mathcal{ O}}(1)$. [ABSTRACT FROM AUTHOR]