Application of Coding Theory in Field 5

Abstract

The main aim of this paper introduce the relationship between the topic of coding theory and the projective plane of order five where special points were found in field 5, which is 31 points, in addition to 31 straight lines, and by applying the theorem that gives the number 1 for the point that lies on the straight line and the number 0 for the point that does not lie on the straight line, we get the code n = 31, d = 6, e = 2, from which we get the table m, v, n. ,n,h, and based on these tables, the distance difference between the code elements was found, where the minimum distance was 6 and the largest distance was 31. These values were used to test the optimality of the code. We can generalize this theorem and apply it to larger fields such as 21 or 23 and others, test their ideality, and find the difference between field 5 and the rest of the fields. M is the maximum value of the size of code over the finite field of order five and an incidence matrix with the parameters, n (length of code), d(minimum distance of code ), and e (error-correcting of code)have been constructed some example and theorem have been given.

Keywords

• projective plane
• coding theory
• incidence matrix
• distance
• perfect code