1. Complete MOLS of Small Finite Fields Abstract: This note explain how to use (p^{n}-1)-MOLS(GF(p^{n})).xls worksheet to compute complete MOLS of Finite Fields GF(p) and GF(p n ). © César Bravo, 2009.

2. Interface

3. Input data To compute the complete MOLS set of Finite Field GF(p n ), the user must provide: A prime number p at cell C1 A prime number p at cell C1 An integer n at cell C2 An integer n at cell C2 OBS: No consistency check is made on p, since a Finite Field’s user must be well aware that there is no Finite Field GF(p n ) when p is NOT prime.

4. (p-1)-MOLS(GF(p)) button Computes Addition table of GF(p) : (GF(p), + ) Addition table of GF(p) : (GF(p), + ) Multiplication table of GF(p): (GF(p), x ) Multiplication table of GF(p): (GF(p), x ) Then applies, as a permutation, each nonzero column of (GF(p), x ) to the rows of (GF(p), + ) to obtain a complete set of (p-1) MOLS.

5. Example: 4-MOLS(GF(5))

6. (p^{n}-1)-MOLS(GF(p^{n})) button Computes The primitive polynomials of GF(p n ) The primitive polynomials of GF(p n ) Addition table of GF(p n ) : (GF(p n ), + ) Addition table of GF(p n ) : (GF(p n ), + ) Multiplication table of GF(p n ): (GF(p n ), x ) Multiplication table of GF(p n ): (GF(p n ), x ) Then, in order to obtain a (p n -1) MOLS(GF(p n )), applies the Theorem [1]: L  (i,j) = (  i) + j, where i, j  GF(p n )+,   GF(p n ).

7. Example: 7-MOLS(GF(2 3 ))

8. “Recompute MOLS” button Recomputes, according to the first irreducible polynomial on Plan4: (GF(p n ), + ) (GF(p n ), + ) (GF(p n ), x ) (GF(p n ), x ) (p n - 1)-MOLS(GF(p n )) (p n - 1)-MOLS(GF(p n )) So, if you need another MOLS set, simply, change the first polynomial on Plan4 to that of your election between those available.

9. Other buttons [2] p(x) root powers: Computes the powers of a root the first polynomial on Plan4 p-adic expansion: Computes the decimal representation of the p-adic expansion of the powers of a root the first polynomial on Plan4. Roots order: Computes the order of the roots of all polynomials on Plan4. Primitive roots: Deletes from Plan4 any polynomial with roots of order different from p n. (GF(p^{n}), +): Computes the addition table of GF(p n ) (GF(p^{n}), x): Computes the multiplication table of GF(p n )

10. References 1. Colbourn, Dinitz. CRC Handbook of Combinatorial Designs. 2. Bravo. Small Finite Fields. Available at: url: http://stoa.usp.br/cesarabp/files/1729 http://stoa.usp.br/cesarabp/files/1729 (This is the original implementation of Finite Field arithmetic upon which this complete MOLS toolbox is based)