Determination of the sizes of optimal geometric orthogonal codes with parameters (n× m,k,λ ,k-1)

The study of (n× m,k,λ _a,λ _c) -geometric orthogonal codes ( (n× m,k,λ _a,λ _c) -GOCs) is motivated by the application in DNA origami. The central research on GOCs is to determine the value of Φ (n× m,k,λ _a,λ _c) , i.e., the largest possible size among all (n× m,k,λ _a,λ _c) -GOCs. When λ _a=λ _c , the exact values of Φ (n× m,3,1,1) and Φ (n× m,k,k-1,k-1) were recently determined by Wang and Su et al. In this paper, we research on the cases of λ _c=k-1 and λ _a ≤ k-2 . We determine the exact values of Φ (n× m,k,k-2,k-1) and Φ (n× m,k,k-3,k-1) , and give a calculation method of Φ (n× m,k,λ _a,k-1) with λ _a≤ k-4 for any positive integers n , m and k .