Vol 55, No 3 (2019)

We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve the estimates existing in the literature. We also present an upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on the metric divergence of an ensemble. In the case of a multi-mode quantum oscillator, this upper bound is tight for large energy. Upper bounds for the Holevo capacity of finite-dimensional quantum channels depending on metric characteristics of the channel output are obtained.

In an earlier paper we developed a unified approach to the extendability problem for arcs in PG(k - 1, q) and, equivalently, for linear codes over finite fields. We defined a special class of arcs called (t mod q)-arcs and proved that the extendabilty of a given arc depends on the structure of a special dual arc, which turns out to be a (t mod q)-arc. In this paper, we investigate the general structure of (t mod q)-arcs. We prove that every such arc is a sum of complements of hyperplanes. Furthermore, we characterize such arcs for small values of t, which in the case t = 2 gives us an alternative proof of the theorem by Maruta on the extendability of codes. This result is geometrically equivalent to the statement that every 2-quasidivisible arc in PG(k - 1, q), q ≥ 5, q odd, is extendable. Finally, we present an application of our approach to the extendability problem for caps in PG(3, q).

Cyclic codes as a subclass of linear codes have practical applications in communication systems, consumer electronics, and data storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to construct some cyclic codes by the sequence approach. More precisely, we determine the dimension and the generator polynomials of three classes of q-ary cyclic codes defined by some sequences with explicit polynomials over \(\mathbb{F}_{q}m\). The minimum distance of such cyclic codes is also discussed. Some of these codes are optimal according to code tables. Moreover, the third class of cyclic codes provides some answers for Open Problem 3 proposed by Ding and Zhou in [1].

Codes with the identifiable “parent” property appeared as one of solutions for the broadcast encryption problem. We propose a new, more general model of such codes, give an overview of known results, and formulate some unsolved problems.

We slightly improve the superconcentrator construction by Alon and Capalbo.

Abstract—We correct mistakes in the formulations of Theorem 19 and Proposition 17 of the original article, published in vol. 55, no. 1, 1–45.