Construction of a fractal using QR code pattern

A subset of Euclidean space is called a fractal if its Hausdorff dimension strictly exceeds its topological dimension. In this paper, we construct a set [Formula: see text] using an iterative scheme inspired by the structural pattern of QR codes. We prove that [Formula: see text] is uncountable, nowhere dense, and has Lebesgue outer measure zero. Furthermore, we show that [Formula: see text] is the attractor of an iterated function system in [Formula: see text]. Without assuming the open set condition, and relying only on the properties of QR code patterns, we compute the Hausdorff dimension of [Formula: see text] and establish that it strictly exceeds its topological dimension.