Page 180 - ISCI’2017

cases, able to correct single errors in SRC d (SRC ) = 2 (at k = 1).
min
Really, if given the expression (3) and (5), for the orderly SRC, it can be draw a conclusion. When

1
one ( k = ) control based m n+ 1 of the SRC, the NCS A = (a 1 || a 2 ||...|| a i− 1 || aa i+ 1 ||...|| a n || a n+ 1 )
||
i
(SRC
may have different meanings d min ) . In this case, it depends on the magnitude of the control base
m n+ 1 .

If, for each individual module of SRC condition m < m n+ 1 (i = 1, n ), then, in accordance with the
i
(SRC
expression (1), we can conclude that d min ) = 2, i. e., in accordance with equation (2) we obtain that
t det. = 1 . If the totality of information bases {}m for an arbitrary pair of modules condition
i
j
(SRC
mm⋅ i j < m ( , ij = 1, n ; i ≠ ), in this case d min ) = 3 и t det. = 2 . Thus, for the NCS in the SRC with
1
n+
(SRC
k = 1, MCD d min ) can be different depending on the magnitude of the control base m n+ 1 of the SRC.
Consider the ratio by which the error can be corrected in the residual a  [1]. Let the wrong number
i


≥

( AM ) A = (а 1 || а 2 ||...|| a i− 1 || aa i+ 1 ||...|| a n || a n+ 1 ) including error a = (a +∆ a i )mod m reliably
||

i
i
i
i
contained in the residue аi modulo mi.
It is obvious

A ( А = А +∆ )mod M 0 . (6)
Given that the amount of error may be represented as A∆ = (0 ||0 ||...||0 || a∆ i ||0 ||...||0 ||0) , when
the correct ( AM< ) number A can be determined as follows:

А= ( – А )mod M = 0  (a 1 || a 2 ||...|| a i− 1 || aa i+ 1 ||...|| a n || a n+ 1 ) (0 ||0 ||...||0 || a i ||0 ||...
−
||
A ∆
∆


i
...||0 || ) 0 mod M = [a 1 || a 2 ||...|| a i− 1 ||(a  i – a∆ i )mod ||ma i+ 1 ||...|| a n || a n+ 1 ]mod M .


i
0
0
Obtain a quantitative estimate of the value of A. Since the number A is correct, i.e. stored in the
numerical range [0, M), then the following inequality must be fulfilled
А ( –A =  А ∆ ) mod M 0 M < . (7)
Given that the value A∆ of the error value is equal А∆ = ∆ aB⋅ i , then the inequality (7) will have
i
the following form:

A −∆ ⋅ i – rM < M or
a B
⋅
i
0

A −∆ a B⋅ i – r M⋅ 0 < M 0 / m n+ 1 (r = 1, 2, 3,...) ,
i
(8)

A –(a  i – a ⋅ i rM < M 0 / m n+ 1 ,
) B −⋅
0
i

A –(a i – a  i ) B⋅ i −⋅ 0 M 0 / m n+ 1 ,
rM <
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