Three bii linear codes of length 27 and 28 are described.They cuntaiu more vectors than any previously known codes with the samelength and miuimIm3 distance. Let gi(x) and g2(x) be the following polynomials over GF(2):

By a careful analysis of the proofs of Kopparty in fields of characteristic 2, we show that the problem of powering in Fpn requires ACC(p) circuits of exponential size (in n), for any fixed prime p > 2. Similar bounds hold for quadratic residuosity. As a corollary, we obtain non-trivial bounds for exponential sums that express the correlation between the quadratic character of Fpn and n-variate...

In this paper we investigate the following question. Let p be a prime, a,, " ', cck distinct non-zero residue classes modp, N a residue class modp. denote Dhe number of solutions of the congruence ela,+... + ekak = N(modp) where the e,,. .. , ek are restricted to the values 0 and 1, What can be said about the function J?(N)? We prove two theorems. THEOREM I. 14 " (X) > 0 if k > 3 (6;~) " ~. THE...

For a fixed rational number g ∈ {−1, 0, 1} and integers a and d we consider the set Ng(a, d) of primes p such that the order of g modulo p is congruent to a (mod d). Under the Generalized Riemann Hypothesis (GRH), it can be shown that the set Ng(a, d) has a natural density δg(a, d). Arithmetical properties of δg(a, d) are described, and δg(a, d) is compared with δ(a, d): the average density of ...

There has been a tendency to use the theory of finite Galois fields, or GF(2n), in cryptographic ciphers (AES, Kuznyechik) and digital signal processing (DSP) systems. It is advisable modular codes polynomial residue number system (PRNS). Modular PRNS are arithmetic which addition, subtraction multiplication operations performed parallel on bases code, irreducible polynomials. In this case, ope...

This paper studies the complexity of learning classes of expressions in propositional logic from equivalence queries and membership queries. In particular, we focus on bounding the number of queries that are required to learn the class ignoring computational complexity. This quantity is known to be captured by a combinatorial measure of concept classes known as the certificate complexity. The p...

A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterized as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and sufficient conditions on sets of forbidden permutations which ensure that the associated pattern class is of polynomial growth is determined. A catalogue all ...