This paper addresses the problem of the mathematical design of a Linear Fresnel Reflector, specifically the design of a reflector concentrator with flat mirrors and a single absorber tube. The mathematical aspects of the design, i.e., the number, width and position of the primary mirrors and the height, length and relative position of the single absorber tube, are analysed. The optimization of the relative position with respect to the primary reflectors and the size of the single absorber tube are both addressed, further analysing up to 12 different configurations. To do so, both the frontal and lateral view of the structure are taken into account. The lateral optical performance factor analysed here, has been overlooked until now, as it may be insignificant in large-scale concentrators. It is shown in this paper that it is a key aspect in medium- and small-scale concentrators, the most common in applications in the Household Sector like, for example, micro-cogeneration. Finally, a number of numerical simulations performed in a custom-designed program compiled using Mathematica (R) is presented. At the time of this writing, a prototype is being built at CIFP in La Felguera, Asturias, Spain. (C) 2016 Elsevier Ltd. All rights reserved.