UNREASONABLE EFFECTIVENESS OF NUMBER THEORY
15
the receiver to distinguish between precisely 23 = 8 different possibilities: a
single error in any of the 7 transmitted bits or no error. No wonder the
Hamming Code is called a perfect code.
9. Correlation and fourier properties of Galois sequences
For many purposes it is advantageous to use the elements sk = 1 or - 1
instead of ak = 0 or 1. The mapping is
Sk =
2a
k
- 1 .
Using this notation, the constant Hamming distance between code words
of a Simplex Code (whose members are generated by cyclic shifts) translates
immediately into the following circular auto-correlation property
n
cr- = £ sksk+r = - 1 for r # 0 mod n ,
k = l
and, of course, cr = n for r = 0 mod n. As a result of this two-valuedness of
cr, the Fourier transform of sk:
n
Sm = Z sk exp(-i27ckm/n)
k=i
has constant magnitude for m # 0 mod n. In the lingo of the physicist and
computer scientist: the sequence sk has a flat (or "white") power spectrum.
If we identify the index k with (discrete) physical time, then we can say
that the "energy"
|skl2
= 1, of the sequence sk is equally distributed over
all time epochs. And because of |S
m
I2 is constant, we can make the same
claim with respect to the distribution of energy over all (non-zero) frequency
components. This equal "energy spreading" of the Galois sequences sk with
period length n =
2m
- 1, obtained with the help of polynomials over
GF(pm),
has many impressive applications, some of the more astounding
ones occurring in the interplanetary distance measurements.
10. Galois sequences and the fourth effect of general relativity
General Relativity, the theory of gravitation propounded by Einstein in
November 1915 in Berlin (and 5 days earlier by Hilbert in Gottingen) passed
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