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where I is the identity submatrix by the size k × k, P is a sub-matrix by the size k × (n – k) in the right

part of the matrix G*, || is a symbol of concatenation (association).
Then, at using of the following expression c = iG, we obtain the systematic code rule c = i || iP,

i.e. the information vector i = (i0, i1, …, ik-1) will be explicitly contained in the code word c = (c0, c1,
…, cn-1).

Linear space that equates code Vk has an orthogonal complement (denote it as Un-k). Basis of
subspace Un-k can be set by the following vectors

(h 0 , 0 ,h 1 , 0 ,...,h , 0 −n 1 ) ,

(h 0 , 1 ,h 1 , 1 ,...,h , 1 −n 1 ),

…
(h n −k − 0 , 1 ,h n −k − 1 , 1 ,...,h n −k − , 1 −n 1 )

and it is usually presented in the matrix kind via a validation matrix

 h h ... h 
 0 , 0 1 , 0 ... , 0 n −1 
, 1 n
H =  h ... 0 , 1 h ... 1 , 1 ... h ... −1 


 
 h h ... h 
 n −k − 0, 1 n −k − 1,1 n −k − ,1 n −1 
of the rank rank (H) = n – k and a dimensionality (n – k) × n.

The condition of the orthogonality vectors from Vk and Un-k in a matrix kind can be write as

GH T = 0, (5)
where “0” is a zero matrix with a dimensionality k × (n – k).

Linear subspace Un-k calls dual (ambivalent) to the code Vk over the field GF(q). Definition of the

code Vk through orthogonal complement Un-k (through dual code) can formulate the following: a
random n-sequences c = (c0, c1, …, cn-1) with the elements from the field GF(q) is a code word of the

T
code Vk if and only if, when it is orthogonal to each row of the validation matrix H, i.e. at cH = 0.
Let the matrix H to the canonical kind by the linear operations on the rows

 h * h * ... h * 1 0 ... 0 
 0 , 0 1 , 0 , 0 k 1 − 
 h * h * ... h * 0 1 ... g * 
H* =  0 , 1 1 , 1 , 1 k 1 − , 1 n 1 −  = P* I , (6)
 ... ... ... ... ... ... ... ... 
 h * h * ... h * 0 0 ... 1 
 n− k 1 − 0 , n− k 1 − 1 , n− k 1 − , k 1 − 

where P* is a submatrix with a dimensionality (n – k) × n in the left part of the matrix H*, I is a single

submatrix with a dimensionality (n – k) × (n – k), || is a symbol of concatenation (association).
Then from the condition G* H* = 0, we have P* = - P (with the operations over the field GF(q)).
T
T

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