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#1 2016-10-24 11:35:33

Seetha Rama Raju Sanapala
Member
Registered: 2016-10-23
Posts: 4

construction of GF(16), its Cayley tables of both the operations, its

I know how to build GF(p) where p is a prime.  It is just simply {0,1,2,  ....p-1) with modulo addition and modulo multiplication p operations.  From the number theory, we can show that the set really satisfies the 11 requirements of a field iff p is a prime.  I also know that similar things happen if we take irreducible polynomials of degree any natural number m over a field - if we take addition and multiplication modulo this polynomial.  So we can take an irreducible polynomial and build a GF(p^m).  We can build the Cayley's tables for these two operations.  Using the tables, we can find the orders of each of the elements and find the primitive roots.  But error control coding books follow a different approach.  They use primitive polynomials and build the field from the root of this 'primitive polynomial'.  I do not know the advantage of following this approach.  And actually I am confused with this approach.  For example, assuming a root for an irreducible polynomial to start with,  whereas root does not exist in the field.  For example existence of root for 1+X+X^3 while considering a binary field.  Can some one help understand the rationale of this approach?

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