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1 X 0 X 0 2 ... X 0 n − 1 T
1 X 1 X 1 2 ... X 1 n − 1
(с 0 ,c 1 ,c 2 ,...,c N − 1 ) ⋅ 1 X 2 X 2 2 ... X 2 n − 1 = 0 .
... ... ... ... ...

1 X r − 1 X r 2 − 1 ... X r n − − 1 1

The resulting expression corresponds to the condition of mutual of the orthogonality random
code word с = (с0, с1, с2,…, сn-1) and a matrix in the right part of multiplication. Consequently, let us

set

 X 0 X 1 ... X n −1 
 0 0 0 1 0 n −1 
cH T = 0, H =  X 1 X 1 ... X 1   , (9)

 ... ... ... ... 
 0 1 n −1 
 X n −k −1 X n −k −1 ... X n −k −1 

where H is a validation matrix of the code, which given by radicals of the generator polynomial.
For building of the matrix H with the elements over the subfield GF(q) should be replaced every

i
element X ∈ GF (q m ) in the expression (9) by the vector-column from m elements of the field
j
GF(q).

If 2t successively following elements X = α j , X = α j+ 1 ,..., X nk 1 = α j+ 21t− ∈ GF (q m ) will
0
1
−−
choose as the radicals of the polynomials g(x), then based on the Bose-Chaudhuri-Hocquenghem’s
theorem [38-40], the resulting code (BCH-code) will have a minimal code distance, which is equal to

d = 2t + 1. Validation matrix will have the following kind for the BCH-code over GF(q )
m
 α 0 α j ... α ( jn− 1) 
 α 0 α j+ 1 ... α ( j+ 1)(n− 1) 
H =   . (10)
 ... ... ... ... 
  1)  
t−
t−
 α 0 α j+ 2 1 ... α ( j+ 2 1)(n− 
Defined in this way codes call Reed–Solomon codes, their (n, k, d) parameters over GF(q )
m
connected by a relation d = n – k + 1 (the high Singleton bound), i.e. these codes have maximum

separable code distance (called MDS-codes) [38-40]. The following codes over GF(q), (n, k, d)-

parameters of which satisfy the constraint (a low BCH-bound) d ≥ n – km + 1 can be obtained with
i
the help of the constraint to the field GF(q) with replacing all α ∈ GF (q m ) in an expression (10)
corresponding column-vectors from m elements of the field GF(q.

Assume that the code word c is distorted during transmission. Let a number of errors on a block

 −1d 
from N symbols does not exceed the ability of correcting =t   of the algebraic (n, k, d)-code.
 2


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