
EXAMPLE

Let g be a generator of the multiplicative group GF(2^6)^* with reduction polynomial t^6+t+1 = 0.
Pick g = t^3+1 (which generator is chosen doesn't matter for the sequence; but it matters for the table below).
Let n be an integer, 0 < n < 2^6  1. Let the smallest positive integer k such that (g^n)^(2^k) = g^n be k = 6, then the elements { g^n, (g^n)^2, (g^n)^(2^2), (g^n)^(2^3), (g^n)^(2^4), (g^n)^(2^5) } are all different and form an FEC with size 6.
These elements are equivalent, any may be chosen as representative.
The inverse of the FEC is an FEC with the inverse of those elements (all of which are in the same FEC of course).
The trace of each element is the same, of course and therefore we might as well speak about the inverse of the FEC and the trace of the FEC respectively.
In the table below, the FEC are denoted as {1,2,4,8,16,32} etc, only giving the exponents of g. All FEC with size 6 are given in both columns, the two columns give each others inverse.
The trace of the FEC is given after the FEC.
.......... x .......... Tr(x) ........ 1/x ........ Tr(1/x)
{ _1, _2, _4, _8, 16, 32} 0 { 31, 47, 55, 59, 61, 62} 1
{ _3, _6, 12, 24, 33, 48} 0 { 15, 30, 39, 51, 57, 60} 1
{ _5, 10, 17, 20, 34, 40} 1 { 23, 29, 43, 46, 53, 58} 1
{ _7, 14, 28, 35, 49, 56} 0 { _7, 14, 28, 35, 49, 56} 0
{ 11, 22, 25, 37, 44, 50} 1 { 13, 19, 26, 38, 41, 52} 0
{ 13, 19, 26, 38, 41, 52} 0 { 11, 22, 25, 37, 44, 50} 1
{ 15, 30, 39, 51, 57, 60} 1 { _3, _6, 12, 24, 33, 48} 0
{ 23, 29, 43, 46, 53, 58} 1 { _5, 10, 17, 20, 34, 40} 1
{ 31, 47, 55, 59, 61, 62} 1 { _1, _2, _4, _8, 16, 32} 0
This shows that there are 3 FEC (namely, 5, 7 and 23) whose trace is equal to the trace of its inverse and hence a(6) = 3.
