A reconfigurable intelligent surface (RIS) is composed of low-cost elements that manipulate the propagation environment from a transmitter by intelligently applying phase shifts to incoming signals before they are reflected. This paper explores a uni-polarized RIS with linear shape aimed at transmitting a common signal to multiple user equipments (UEs) spread across a wide angular region. To achieve uniform coverage, the uni-polarized RIS is designed to emit a broad and spectral-efficient beam featuring a spatially flat-like array factor, diverging from the conventional narrow beam approach. To achieve this objective, we start by deriving probabilistic lower and upper bounds for the average spectral efficiency (SE) delivered to the UEs. Leveraging the insights from the lower bound, we focus on optimizing the minimum value of the power domain array factor (PDAF) across a range of azimuth angles from \(-\fracπ{2}\) to \(\fracπ{2}\). We employ the continuous genetic algorithm (CGA) for this optimization task, aiming to improve the SE delivered to the UEs while also creating a wide beam. Extensive simulation experiments are carried out to assess the performance of the proposed code, focusing on key metrics such as the minimum and average values of the PDAF and the SE delivered to the UEs. Our findings demonstrate that the proposed code enhances the minimum SE delivered to the UEs while maintaining the desired attribute of a broad beam. This performance is notably superior to that of established codes, including the Barker, Frank, and Chu codes.