Page 181 - ISCI’2017


( a i – a  i ) B⋅ i < M 0 / m n+ 1 – A rM+ ⋅ 0 ,

a i – a < (M 0 / m n+ 1 )/ B i – A B + r M⋅ 0 / B ,
/

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i

/
a +
a <  i (M 0 / m n+ 1 )/ B i – A B + r M⋅ 0 / B .
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Given that the orthogonal basis for the module m of the SRC is represented as B = mM⋅ / m ,
i i i 0 i
the expression (8) becomes:

/
a <  i (m + ⋅ i ⋅ n+ 1 )/ (mm n+ 1 ) – A B or
⋅
a +
r mm
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i

/
a <  i m i (1 rm+ ⋅ n+ 1 ) / (m m⋅ n+ 1 ) – A B . (9)
a +
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i
Since the value of the residual ai is a natural number, then the value of

/
m i 1 ( + ⋅ n+ 1 / )(m m ⋅ n+ 1 ) – A B in the expression (9) must be an integer. Therefore, taking the
rm
i
i

whole of the last relation, we obtain a formula for the correction of an error in the residual a  of A
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as

]
/
+ ⋅
а = i (a + [m ⋅ i (1 rm n+ 1 ) (/ m ⋅ i m n+ 1 ) – A B i ) mod m . (10)

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To confirm the results of theoretical studies, we consider examples of monitoring and correction
of the data in the SRC.
Example 1. Implement control and, if necessary, to carry out a correction of the number
4
5
A SRC = (0 ||0 ||0 ||0 ||5) which is set in the SRC with the information base m = 3, m = , m = ,
2
3
1
n 4
i ∏
m = 7 and with the reference base m = m = 11 . However M = ∏ m = m = 420 and
k
5
i
5
i= 1 i= 1
⋅
=
⋅
M = 0 Mm n+ 1 = 420 11 4620. Orthogonal bases B (i = 1, n + 1) of SRC are in [6].
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I. Data control A SRC = (0 ||0 ||0 ||0 ||5). In accordance with the control procedure [1], we define the
value of
 n+ 1   5 
⋅
⋅
A PNS =  aB i ∑ mod M = 0  aB i ∑ mod M = 0 (a B + 1 ⋅ 1 a B + 2 ⋅ 2 a B + 3 ⋅ 3
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i
 i= 1   i= 1 
a ⋅ B + aB⋅ 5 )mod M = (0 1540 0 3465 0 3696 0 2640 5 2520)mod 4620⋅ + ⋅ +⋅ +⋅ + ⋅ =
4
5
0
4
= (5 2520)mod 4620 12600(mod 4620)⋅ = = 3360 > 420.
Thus, the process control is determined that A SRC = 3360 > M = 420 . In this case, the possible

occurrence of only once errors, it is concluded that the number of considered A 3360 = (0 ||0 ||0 ||0 ||5)

is incorrect (3360 M> = 420). To correct the number A 3360 = (0 ||0 ||0 ||0 ||5), you must first make a
diagnosis data, i.e. identify distorted residual a  . Then it is necessary to determine the true value of
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the residual a to modular m and then spend correcting distorted residual a  .
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