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increase by 2 symbols with the help of modification code procedures). With the help of a restriction

to the subfield GF(q) can obtain a large length n-code at the fixed q, but the (n, k, d)-code parameters
are lying significantly below code borders (1) and (2), and this tendency increases at an increasing of

n- length. Nevertheless, some algebraic code classes are exist, which are lying above code borders
(1) and (2).

m
Definition 1 [38-40]. Let Х = (Х0, Х1, …, Хn-1) vector over the field GF(q ), at that all Хi –
m
m
different elements GF(q ). Let also B = (B0, B1, …, Bn-1) is a vector over the field GF(q ) with
m
optional and different Bi elements of the field GF(q ). Then (n, k, d) generalized Reed-Solomon code
ОРСk(Х, h) consists from all vectors of kind
(B0∙F(Х0), B1∙F(Х1), …, Bn-1∙F(Хn-1)),

where F(x) is any polynomial with the coefficients from the field GF(q ), a degree of which does not
m
exceed k. ОРС-code is MDS-code, its validation matrix ОРСk(Х, h) is equal:

 Y 0 Y 1 ... Y n− 1 
 ... 
 XY ⋅ 1 0 X Y ⋅ 2 1 X n− 1 Y ⋅ n− 1 
H =  X ⋅ 2 Y X ⋅ 2 Y ... X 2 Y ⋅  =
 1 0 2 1 n− 1 n− 1 
 ... ... ... ... 
 nk 1 nk 1 nk 1 
−−
−−
−−
 X 1 Y ⋅ 0 X 2 Y ⋅ 1 ... X n− 1 Y ⋅ n− 1  (15)
 1 1 ... 1  Y 0 0 ... 0 
 X X ... X  0 Y ... 0 
 1 2 n− 1  1 
=  X 2 X 2 ... X 2  ⋅ 0 0 ... 0  ,
 1 2 n− 1  
 ... ... ... ...  ... ... ... ... 
 nk 1 nk 1 nk 1  
−−
−−
−−
 X 1 X 2 ... X n− 1  0 0 ... Y n− 1 
where a vector Y = (Y0, Y1, …, Yn-1) such that ∀Yi ∈ GF(q ), Yi ≠ 0 and ОРСn-k(Х, Y) is dual to
m
ОРСk(Х, B).
Via definition ОРС let us introduce an extensive class the so-called alternantive codes [38-40].

Definition 2 [38-40]. Alternantive (n, k, d)-code A(X, B) consists of all the code words
ОРСk(Х, B) such that their components are lying if the field GF(q). In other words, A(X, B) equals a

restriction of the code ОРСk(Х, B) to the subfield GF(q), i.e. its consists of all vectors c over GF(q),

for which the expression сH T = 0 is executed, where H is the validation matrix ОРСk(Х, B), defined
by the equation (15). Generating matrix A(X, B) can be obtained by replacing of each element of the

matrix H in (15) by the corresponding column-vector of length m over GF(q).

Code parameters A(X, B) related by the formula: n – mr ≤ k ≤ n – r; d ≥ r + 1, at that it was
proved [38-40], that among a large number of alternantive possible codes at fixed n and k be found

such codes, parameters of which are lying above code boundaries (1) and (2). One of the special cases

A(X, B) are Goppa codes [42, 43].

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