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