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We propose several constructions of nonbinary constant-weight codes meeting the Johnson bound. [ABSTRACT FROM AUTHOR]

We present two new families of resolvable block designs. We introduce the notion of a weakly resolvable block design and prove the equivalence of such designs and nonbinary codes meeting the Johnson bound. We construct a new family of such codes. [ABSTRACT FROM AUTHOR]

We prove that any symmetric block design (v, k, λ) generates optimal ternary and quaternary constant-weight equidistant codes, whose parameters n, N, w, d, q are uniquely determined by the parameters of the block design. For one rather special case, we construct symbolwise uniform equidistant codes of the minimum length. [ABSTRACT FROM AUTHOR]

We introduce m-near-resolvable block designs. We establish a correspondence between such block designs and a subclass of (optimal equidistant) q-ary constant-weight codes meeting the Johnson bound. We present constructions of m-near-resolvable block designs, in particular based on Steiner systems and super-simple t-designs. [ABSTRACT FROM AUTHOR]

We prove that any balanced incomplete block design B( v, k, 1) generates a nearresolvable balanced incomplete block design NRB( v, k − 1, k − 2). We establish a one-to-one correspondence between near-resolvable block designs NRB( v, k −1, k −2) and the subclass of nonbinary (optimal, equidistant) constant-weight codes meeting the generalized Johnson bound. [ABSTRACT FROM AUTHOR]

We slightly improve the superconcentrator construction by Alon and Capalbo. [ABSTRACT FROM AUTHOR]

A method for constructing optimal almost equisymbol codes based on difference matrices is proposed. Some of these codes generate previously unknown optimal equidistant codes satisfying the Plotkin bound. As a consequence, they generate new resolvable balanced incomplete block designs. [ABSTRACT FROM AUTHOR]

We derive an asymptotic upper bound on the decoding failure probability for an MDS code in a q-ary multiple access channel. [ABSTRACT FROM AUTHOR]

We study the divisibility by eight of exponential sums of several classes of functions over finite fields of characteristic two. For the binary classical Kloosterman sums K( a) over such fields we give a simple recurrent algorithm for finding the largest k, such that 2 divides the Kloosterman sum K( a). This gives a simple description of zeros of such Kloosterman sums. [ABSTRACT FROM AUTHOR]

We give explicit constructions of polynomials of special form over finite fields the sum of whose multiplicative characters is exactly known. [ABSTRACT FROM AUTHOR]

We present an explicit construction for polynomials of special form over a finite field whose trigonometric sum is exactly known. [ABSTRACT FROM AUTHOR]

The Grey-Rankin bound for nonbinary codes is obtained. Examples of codes meeting this bound are given. [ABSTRACT FROM AUTHOR]

An explicit construction of polynomials over a finite field of odd characteristic, for which the absolute value of trigonometric sums attains the Weil bound, is based on the construction of cyclic matrices of given rank. Dickson polynomials of the second kind play an essential role in the study of such matrices. [ABSTRACT FROM AUTHOR]

We study trigonometric sums in finite fields $$F_Q $$ . The Weil estimate of such sums is well known: $$|S(f)| \leqslant ({\text{deg }}f - 1)\sqrt Q $$ , where f is a polynomial with coefficients from F(Q). We construct two classes of polynomials f, $$(Q,2) = 2$$ , for which $$|S(f)|$$ attains the largest possible value and, in particular, $$|S(f)| = ({\text{deg }}f - 1)\sqrt Q $$ . [ABSTRACT FROM AUTHOR]

A theorem of Lloyd is extended to the Lie metric case. [ABSTRACT FROM AUTHOR]

This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1134/S0032946020040092. [ABSTRACT FROM AUTHOR]

The acknowledgment (footnote to the title of the paper) should read as follows: The research was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50- 00150. [ABSTRACT FROM AUTHOR]

This note is a comment to [1], where it was proved that, over a field of odd characteristic, the number of square matrices of order n, n ≥ 3, with nonzero permanent is greater than the number of square matrices of the same order with nonzero determinant. [ABSTRACT FROM AUTHOR]