This paper introduces a new nonclassical statistical distribution, SMART1, derived from a nonpolynomial function. This function is used in ecology to model population growth rates, in medicine to classify the relationship between activity and tumor volume in cancer, and in economics to analyze the relationship between supply and demand. The proposed distribution differs from the Gompertz distribution, which is fundamentally based on the exponential function. By contrast, the SMART1 distribution is constructed on principles of mathematical analysis, specifically through the identification of local maximum endpoints of the Gompertz growth function and the subsequent verification that the resulting function satisfies the criteria of a probability density function. This distribution (SMART1) is flexible and can model various real-world phenomena on a bounded interval (0, β ), especially in reliability analysis or survival modeling, where an upper lifetime limit exists. All statistical concepts are expressed in terms of the distribution's scale parameter β defines the upper bound (or support limit) of the distribution. The random variable X can only take values in the interval (0, β ). In other words, β is a ``lifetime limit'' or ``maximum capacity''; and the shape parameter α determines how the risk or likelihood is distributed over time: Low α : high initial risk that decreases (e.g., early failures). High α : low initial risk that increases with time (e.g., aging or wear-out). These include the probability density function, the cumulative distribution function, the reliability function, the hazard function, order statistics, moments, and key measures such as the mode and the median.

A k-arcs is usually defined to be a set of k points in the projective plane such that some lines meets K in two points. A conic is an irreducible plane quadric curve with six terms. The first aim of the paper is to find the projectively inequivalent 5-arcs and 6-arcs in the finite projective plane over the Galois field of order two, and find the subgroups of PGL( 3,32 ) that are fixing these arcs. The second aim is to determine the conics form that passes through these 5-arcs and 6-arcs, and then parameterized. As an application to our results over Galois field of order two, the connection between projective linear code and arc in the finite projective plane was taken advantage determine the number of non-equivalent projective MDS linear codes of length five, dimension three and one error correcting. Also, the dual codes of these codes are introduced. Finally, the weight distribution, covering radius and number of correcting of these projective MDS linear codes and its duals are computed. The GAP programming tools were used to implement the algorithms used in this research.