EN|RU Volume 17, No 5, 2010, P. 3-14 UDC 519.725 Ts. Ch.-D. Batueva A family of two-dimensional words with maximal pattern complexity 2k Abstract: Maximal pattern complexity $p^*(k)$ is one of the counting functions over infinite words. In this paper we consider it over two-dimensional words. We construct an infinite family of two-dimensional words with the maximal pattern complexity $p^*(k)=2k$ for $k\in\mathbb N$. It is the minimum of maximal pattern complexity over two-dimensional and not two-periodic words. Bibliogr. 21. Keywords: complexity, maximal pattern complexity, two-dimensional word, Toeplitz word. Batueva Tsyndyma Chimit-Dordjievna 1 1. Novosibirsk State University, 2 Pirogov St., 630090 Novosibirsk, Russia e-mail: cendema@ngs.ru © Sobolev Institute of Mathematics, 2015