Page 67 - ISCI’2017

If it is assumed that the information sequence will be converted into all possible vectors e of n-

length and the (w(e) ≤ t)-weight, then the last expression becomes:

 t  n!  
∑ log q  ( q − )1 i i (! i)!   
*
R = i  =0  n −   . (25)
n − k
Algorithm of information sequence coding in the equilibrium sequence e of n-length and the
w(e)-weight for an arbitrary base q presents, for example, in the work [52].

The expression (25 reaches its maximum for the so-called perfect codes, (n, k, d)-code
parameters of which satisfy the upper Hamming boundary of the power (number of code words)

A q (n , ) d arbitrary linear q-ary code [38 – 40]:

q n
A ( n, d) ≤ t . (26)
q
∑ q ( − )1 i n!
= i 0 i! n ( − i)!
Power of linear (n, k, d)-code over the field GF(q) is equal to q , consequently, from the
k
expression (26) follows a restriction on the number of data code symbols

 t n! 
k ≤ n − log q ∑ q ( − )1 i  .


 =i 0 i! n ( − i)! 
If the parameters of (n, k, d)-code are satisfy the upper the Hamming boundary, i.e. an equality
is reached in the expression (26) and the code is perfect, then


t

log q ∑ ( q − )1 i n!  = n − k , (27)


i  =0 i (! n − i)! 
that after a substitution in the expression (25) gives R = 1, i.e. the relative transmission rate is
*
maximum and a cryptogram in the Niederreiter scheme does not contain redundant symbols.
Let us show the perfect binary Hamming’s code (q = 2) as an example, correcting one error (t = 1).

m
m
This code was defined for any positive integer m > 2 and has the following code parameters (2 -1,2
– m – 1,3) [38 – 40]. Obviously that for these values

t
∑ n! = 2 , n − k = m
m
i 0= i! n ( − i)!
and the relative speed (25) is equal to 1.
Another example is the perfect binary Golay’s code with the parameters (23, 12, 7) [38 – 40].

This code allows to correct (t = 3) errors and for this values we have

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