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- code polynomial (xc ) = c 0 x 0 + xc 1 1 + ...+ c n − 1 x n − 1 (corresponds to the vector c);

- error polynomial (xe ) = e 0 x 0 + xe 1 1 + ...+ e n − 1 x n − 1 (corresponds to the vector e);

0
1
... c
- code polynomial with errors *( )c x = c * x + c * x ++ * n− 1 x n− 1 (corresponds to the vector c );
*
0
1
- syndrome polynomial (xs ) = s 0 x 0 + xs 1 1 + ...+ s n −k − 1 x n −k − 1 (corresponds to the vector s).
,
We define, for example, with the help of the radicals X X 1 ,..., X nk 1 ∈ GF (q m ) , reduced non-
0
−−
g
zero polynomial (x ) = ( − Xx 0 )( − Xx 1 )...( − Xx n −k 1 − ) of a degree r = n – k and the coding rule
c (x ) i= (x )g (x ), (8)
which is an analog of polynomial expressions (3).

Polynomial g(x) by analogy with the matrix G call generative, and the corresponding to its matrix
G can be cyclically obtained by interlaced shifted record of coefficients of the polynomial g(x) [38-

40]. Linear block codes defined so call cyclical, because a utensil of any cyclically shifted sequence

i
(that interpreted as a polynomial x c ()x , i∈ 0,1,...,nk 1 with the operations in a ring in modulus
−−
binomial x – 1) also follows from a utensil to the space Vk of a sequence c (and the corresponding
n
polynomial c(x)).

m
Polynomial g(x) in a general case defined over the GF(q ) and then the code polynomials c(x) also
will be to define over the extended field GF(q ). However, in a case, when all radicals
m
X 0 , X 1 ,..., X nk−− 1 ∈ GF (q m ) are all radicals of some set of minimal polynomials of the elements

m
GF(q ), then the generating polynomial g(x) always will be to have the coefficients from the subfield
GF(q), at that:

 
∏
. . .
gx Н ОК  f i () x ,
() =
 i 
where the variable i runs through all classes of conjugate elements of the field GF(q ), ()fx is a
m
i
i
minimal polynomial of the element α ∈ GF (q m ), α is a primitive element, Н .. .ОК is least common
multiple.
The value of the polynomial in its radical is equal to zero, i.e. for all X ∈ {X , X ,..., X } we
j 0 1 n −k − 1
have the equality

c (X j ) = c 0 X 0 j + Xc 1 1 j + ...+ c n − 1 X n j − 1 ,

which corresponds to recording in matrix form:

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