Adv Syst Sci Appl 2023; 04:31–39
Published online at https://ijassa.ipu.ru.

Study of a Redundant Residue Number System
for Single Error Correction

Viktor Kuchukov1*, Mikhail Babenko2, Safwat Al-Galda3

1North-Caucasus Center for Mathematical Research, North-Caucasus Federal University, Stavropol, Russia
2North-Caucasus Federal University, Stavropol, Russia

3University of Misan, Amarah, Iraq

Abstract: One approach to reliability is to use a redundant residue number system. In general,
two redundant moduli are required to detect and correct a single error. This paper considers an
approach to error correction using a single redundant modulo, which allows a significant reduction
in the hardware used, but at a significant cost in computational speed. The use of an approximate
method based on the Chinese remainder theorem allows the speed of computation to be increased
by eliminating the computationally complex operation of taking the remainder from the division
by the range of the residue number system. A method based on the approximate method with
one redundant modulo is proposed. Modelling of the considered methods on ASICs in RTL and
physical synthesis environment Cadence Genus Synthesis Solution is carried out.

Keywords: residue number system, error correction, Chinese remainder theorem, application-
specific integrated circuit

1. INTRODUCTION

The problem of reliability and guaranteed validity of the results obtained is relevant for
remote control systems where autonomy and fault tolerance are important, e.g. on railways
and oil and gas pipelines. It is known that for data transmission channels the probability of
single errors significantly exceeds the probability of errors of higher multiplicity (generalised
binomial law of error distribution), and for digital devices the law of error distribution is close
to uniform [1, 2]. Most correction codes, especially block binary codes, which include parity
check codes, have two groups of digits — information and control digits. The information
group contains digits that represent the numerical value of the coded value, and the control
group contains digits that are redundant and additionally introduced for the purpose of
detecting and correcting possible distortions [3].

Since there is no possibility of composing the control part of the result by the control
parts of the arithmetic operation components, the possibility of controlling the correctness
of arithmetic operations is excluded. This non-arithmetic character of special position codes
prevents their use in calculating machines, for which the control of arithmetic operations is of
particular importance. One of the examples of arithmetic codes are AN codes [4,5], but these
codes allow only addition and subtraction operations. Another approach to the construction
of arithmetic codes is the residue number system (RNS).

Further, the paper is organized as follows. Section 2 considers a redundant residue number
system and shows the basic properties of error correction in RNS. Section 3 considers the

∗Corresponding author: vkuchukov@ncfu.ru



32 V. KUCHUKOV, M. BABENKO, S. AL-GALDA

residue number system with one redundant modulo for single error correction. Section 4
shows the application of the approximate method based on the Chinese remainder theorem
for error correction in RNS. Section 5 is devoted to the application of the approximate method
to RNS with one redundant modulo. Section 6 presents the modelling of the considered
methods.

2. REDUNDANT RESIDUE NUMBER SYSTEM

One of the most promising representations of numbers in parallel processing is the use of
non-positional number systems, such as the residue number system.

If a set of positive integers p1, p2, . . . , pn, called moduli of the system, is given, then
a residue number system is a system in which a positive integer is represented as a set
of residues on the chosen moduli X = (x1, x2, . . . , xn), where xi = X mod pi for i =
1, 2, . . . , n [3]. If the moduli satisfy the condition p1 < p2 < . . . < pn, then the system is
called ordered.

It is known from number theory that if the moduli pi are mutually prime, then the
representation of the number X = (x1, x2, . . . , xn) < P = p1 · p2 · . . . · pn is singular, where
P is the dynamic range of the number representation. The peculiarity of the residue number
system is the possibility of performing addition, subtraction and multiplication operations in
parallel and independently for each of the modulo [6].

If we add to the residue number system with moduli {p1, p2, . . . , pn} and dynamic range
P =

∏n
i=1 pi, which is often called the working range, an additional modulo pn+1 > pi,

i = 1, n, then the full range of the system is P = P · pn+1. In the case of operations on
numbers in the range [0, P ), if the result of the operation is less than P , it is correct,
otherwise it is incorrect. Thus, the introduction of the redundant modulo pn+1 allows us to
detect calculation errors. The introduction of constraints on the redundant modulo or the
introduction of additional moduli allows not only the detection but also the correction of
errors [7, 8].

Let’s take as an example the RNS {3, 5, 7, 8}, where p4 = 8 is the control modulo. Then
the working range P = 105, the full range P = 840. Take the number X = (1, 2, 3, 4) = 52,
since X < P , there is no error. Suppose there is an error on the second modulo and the
number X = (1, 3, 3, 4) = 388, and since X > P , the error can be detected.

One of the methods of locating errors in modular code is the method of projections. The
projection Xi of a number X = (x1, x2, . . . , xn+1) onto the modulo pi is the number obtained
by crossing out the digit xi in the representation of X .

If in an ordered residue number system the projection Xi of a number X =
(x1, x2, . . . , xi, . . . , xn, xn+1) on the modulo of pi satisfies the condition

Xi >
P

pn+1

,

then the digit xi is correct if only one error is possible [3]. The introduction of only one
control modulo generally fails to locate the error. To correct the error, we can use a method
based on the Chinese remainder theorem (CRT), by which the number X can be obtained
from the formula

X =

∣∣∣∣∣
n∑

i=1

Pi · xi ·
∣∣P−1

i

∣∣
pi

∣∣∣∣∣
P

, (2.1)

where P is the dynamic range, Pi = P/pi,
∣∣P−1

i

∣∣
pi

is the multiplicative inversion of Pi

modulo pi, and the operator |X|pi denotes the remainder of the division of X by pi, i.e.
X mod pi [9].

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



STUDY OF A RRNS FOR SINGLE ERROR CORRECTION 33

Consider the above number X = (1, 3, 3, 4) = 388, for which the projections are X1 =
(3, 3, 4) = 108, X2 = (1, 3, 4) = 52, X3 = (1, 3, 4) = 28, X4 = (1, 3, 3) = 73. We can see
that the projections X2, X3, X4 are within the working range and the error cannot be
corrected.

Let’s take a RNS with two redundant moduli {3, 5, 7, 11, 13} with working range P = 105
and full range P = 15015. Let’s introduce an error in the number X = (1, 2, 3, 8, 0) = 52 on
the second modulo X = (1, 3, 3, 8, 0) = 6058. Then the projections X1 = 1053, X2 = 52,
X3 = 1768, X4 = 598, X5 = 283. From here we can see that only the projection X2 falls
within the working range and then the remainder of X2 = 52 modulo p2 = 5 is 2 and the
corrected number X = (1, 2, 3, 8, 0).

3. CORRECTION OF AN ERROR WITH ONE REDUNDANT MODULO

The paper [10] proposes an approach to error correction with one redundant modulo. Given a
residue number system with a control modulo p1 < p2 < p3 < . . . < pn < pn+1, it is assumed
that the control modulo is reliable and cannot contain an error. If pn+1 > pn · pn−1, then
algorithm 1 is executed.

Algorithm 1. Number recovery based on the method of projections and CRT

Require: X ′ =
(
x′
1, x

′
2, . . . , x

′
n, x

′
n+1

)
Ensure: X

Data in memory: {p1, p2, . . . , pn+1}, P =
∏n

i=1 pi, P = pn · P
wi =

∣∣∣P−1

i

∣∣∣
pi
· P i, where P i = P/pi, for all i ∈ [1, n+ 1]

wi,j =
∣∣∣P−1

i,j

∣∣∣
pj
· P i,j , where P i,j = P i/pj , for all i ∈ [1, n+ 1] and j ̸= i

1: S = 0
2: for i = 1 to n+ 1 do
3: S = S + x′

i · wi

4: end for
5: S = S mod P
6: if S < P then
7: X = S
8: return X
9: else

10: for i = 1 to n+ 1 do
11: Si = 0
12: for j = 1 to n+ 1 do
13: if i ̸= j then
14: Si = Si + x′

i · wi,j

15: end if
16: end for
17: X = Si mod P i

18: if X < P then
19: return X
20: end if
21: end for
22: end if

This algorithm allows a working modulo error to be corrected. However, the error in
the control modulo is not always corrected exactly. Let’s take the RNS {3, 5, 7, 37} and

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



34 V. KUCHUKOV, M. BABENKO, S. AL-GALDA

the number X = (1, 2, 3, 15) = 52 and introduce the error on the control modulo X =
(1, 2, 3, 17) = 1312. Algorithm 1 gives the number X ′ = (2, 2, 3, 17) = 17.

However, the number of projections required is reduced if the reliability of the calculations
on the control modulo is ensured.

4. APPROXIMATE METHOD BASED
ON THE CHINESE REMAINDER THEOREM

Formula (2.1) and Algorithm 1 use the operation of finding the remainder of the division
by a large modulo P. This operation is computationally expensive. One way to improve the
efficiency of this operation is to use an approximate method based on CRT.

To detect and correct a single error in the RNS, we consider the addition of two redundant
moduli pn+1 and pn+2. In [11, 12] a fractional approximate representation of numbers based
on CRT is proposed. Let us divide (2.1) by P and we get

X

P
=

∣∣∣∣∣∣∣
n+2∑
i=1

xi ·

∣∣∣P−1

i

∣∣∣
pi

pi

∣∣∣∣∣∣∣
1

=

∣∣∣∣∣
n+2∑
i=1

xi · ki

∣∣∣∣∣
1

, (4.2)

where ki =

∣∣∣P−1
i

∣∣∣
pi

pi
are constants of the chosen system, and the operation |x|1 means taking the

fractional part of the number x. In this case, the value of the expression (4.2) is in the interval
[0, 1). Taking the fractional part of a number is much simpler than finding the remainder of
the division by the full range, but in hardware implementation the coefficients ki can rarely
be represented as finite fractions and there is a question of rounding accuracy.

To solve this problem, the fractional coefficients ki are multiplied by 2N , where N is the
number of binary digits after the decimal point, providing the necessary level of accuracy of
the calculations, each number obtained is rounded up to an integer, and then all calculations
are performed modulo 2N . Finding the remainder by this modulo is solved in hardware by
truncation, which is a trivial task. For the calculations we can use the estimation proposed
in [13]:

N =

⌈
log2

(
P ·

n+2∑
i=1

(pi − 1)

)⌉
.

In this case, the working range P is represented in the redundant RNS and it is obvious that
P = (0, . . . , 0, πn+1, πn+2), where πn+1 = P mod pn+1, πn+2 = P mod pn+2. Thus, when
calculating projections, it is necessary to find relative values for both X and P .

Let’s look at a similar example in RNS with two redundant moduli {3, 5, 7, 11, 13} and
the number X = (1, 3, 3, 8, 0) = 6058. According to formula (4.2), the coefficients ki are
k1 = 1/3, k2 = 2/5, k3 = 5/7, k4 = 1/11, k5 = 6/13, rounded to the nearest N = 19. To
reduce the record, we will use fractional representation.

Then the relative value for X is 466/1155 and for P is 1/143. Since 466/1155 > 1/143,
the number X contains an error.

For projection modulo p1, the coefficients ki are k1 = 1/5, k2 = 1/7, k3 = 3/11, k4 =
5/13. Then the relative value for X1 is 81/385 and for P is 3/143. Since 81/385 > 3/143,
the number X1 contains an error. For projection modulo p2, the coefficients ki are k1 = 2/3,
k2 = 4/7, k3 = 5/11, k4 = 4/13. Then the relative value for X2 is 4/231 and for P is 5/143.
Since 4/231 < 5/143, the number X2 is error free. The other projections show that the
number contains an error.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



STUDY OF A RRNS FOR SINGLE ERROR CORRECTION 35

To recover the number we need to multiply the value of 4/231 by the full range of the
projection, i.e. by P/p2 = 3003, we get 52, which corresponds to the number without error.

5. APPLICATION OF THE APPROXIMATE METHOD TO ERROR CORRECTION
WITH ONE REDUNDANT MODULO

Let us consider the application of the approximate method to Algorithm 1. The disadvantage
of this algorithm is the need to find the remainder by the large modulo. Let us introduce
Algorithm 2, which allows us to correct the working modulo error in RNS with one redundant
modulo pn+1 > pn · pn−1.

Algorithm 2. Number recovery based on the projection method and approximate CRT

Require: X ′ =
(
x′
1, x

′
2, . . . , x

′
n, x

′
n+1

)
Ensure: X

Data in memory: {p1, p2, . . . , pn+1}, P =
∏n

i=1 pi,
P = pn · P , π = P mod pn+1

P i = P/pi, wi =

∣∣∣P−1
i

∣∣∣
pi

pi
, i ∈ [1, n+ 1]

P i,j = P i/pj , wi,j =

∣∣∣P−1
i,j

∣∣∣
pj

pj
, i ∈ [1, n], j ∈ [1, n+ 1], i ̸= j

1: X =
∣∣∑n+1

i=1 x′
i · wi

∣∣
1

2: W = |π · wn+1|1
3: if X < W then
4: return X · P
5: else
6: for i = 1 to n do
7: Xi =

∣∣∣∑n+1
j=1,j ̸=i x

′
j · wi,j

∣∣∣
1

8: Wi = |π · wi,n+1|1
9: if Xi < Wi then

10: return Xi · P i

11: end if
12: end for
13: end if

Let’s look at an example of how Algorithm 2 works for RNS {3, 5, 7, 37} and the number
X = (1, 2, 3, 17) = 1312.

The data stored in memory:

P = 105, P = 3885, π = 31,
P1 = 1295, P2 = 777, P3 = 555, P4 = 105,
P1,2 = P2,1 = 259, P1,3 = P3,1 = 185, P1,4 = P4,1 = 35, P2,3 = P3,2 = 111,
P2,4 = P4,2 = 21, P3,4 = P4,3 = 15,

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



36 V. KUCHUKOV, M. BABENKO, S. AL-GALDA

w1 =
2

3
, w2 =

3

5
, w3 =

4

7
, w4 =

6

37
,

w1,2 =
4

5
, w1,3 =

5

7
, w1,4 =

18

37
,

w2,1 =
1

3
, w2,3 =

6

7
, w2,4 =

30

37
,

w3,1 =
2

3
, w3,2 =

1

5
, w3,4 =

5

37
.

Then

X =

∣∣∣∣1 · 23 + 3 · 3
5
+ 3 · 4

7
+ 15 · 6

37

∣∣∣∣
1

=
2383

3885
, W =

∣∣∣∣31 · 6

37

∣∣∣∣
1

=
1

37
.

Since 2383/3885 > 1/37, the number contains an error. Let’s construct the projections.
Then the first projection

X1 =

∣∣∣∣3 · 45 + 3 · 5
7
+ 15 · 18

37

∣∣∣∣
1

=
1088

1295
, W1 =

∣∣∣∣31 · 1837
∣∣∣∣
1

=
3

37
.

Since 1088/3885 > 3/37, the number contains an error. The second projection

X2 =

∣∣∣∣1 · 13 + 3 · 6
7
+ 15 · 30

37

∣∣∣∣
1

=
52

777
, W2 =

∣∣∣∣31 · 3037
∣∣∣∣
1

=
5

37
.

Since 52/777 < 5/37, the number does not contain an error. The third projection

X3 =

∣∣∣∣1 · 23 + 3 · 1
5
+ 15 · 5

37

∣∣∣∣
1

=
373

555
, W3 =

∣∣∣∣31 · 5

37

∣∣∣∣
1

=
7

37
.

Since 373/555 > 7/37, the number contains an error.
So the error was in the second modulo. To recover the number, it is necessary to multiply

52/777 by P2 = 777, so the correct number is 52. This approach can also contain an error
only on working moduli.

6. MODELLING

Single error correction simulations were performed on an ASIC in the RTL and physical
synthesis Cadence Genus Synthesis Solution environment using the osu018 stdcells library.
The metrics measured were the time for the signal to travel through the circuit (picoseconds,
ps), the power required (watts, W) and the area used (square microns, µm2).

RNS sets with one and two redundant moduli with 4-6 working moduli covering the 8,
16, 24 and 32 bit ranges shown in Table 6.1 are selected for the simulation.

Tables 6.2–6.4 show the simulation results for the signal propagation time through the
circuit (picoseconds, ps), the area used (square microns, µm2) and the power required (watts,
W). Algorithm 1 with one redundant modulo, denoted as I in the tables, an approximate
method based on CRT with two redundant moduli, denoted as II, an approximate method
based on CRT with one redundant modulo, denoted as III, are chosen for the simulations.

Tables 6.2–6.4 show that the approximate method with two moduli is on average 0.43%
faster, uses 16.96% more area and has a 25.44% higher power consumption compared to the
approximate method with a one redundant modulo. The single redundant modulo method is
on average 180% slower, uses 23% more power but 69% less area.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



STUDY OF A RRNS FOR SINGLE ERROR CORRECTION 37

Table 6.1. Sets of moduli for modelling

Number of
working
moduli

Working range size, bits

8 16 24 32

One
redun-
dant
modulo

4 {3, 5, 7, 11, 79} {13, 17, 19, 23,
439}

{59, 61, 67, 71,
4759}

{251, 257, 263,
269, 70753}

5 {2, 3, 5, 7, 11,
79}

{5, 7, 11, 13, 17,
223}

{23, 29, 31, 37,
41, 1523}

{79, 83, 89, 97,
101, 9803}

6 {2, 3, 5, 7, 11,
13, 149}

{3, 5, 7, 11, 13,
17, 223}

{11, 13, 17, 19,
23, 29, 673}

{31, 37, 41, 43,
47, 53, 2503}

Two
redun-
dant
moduli

4 {3, 5, 7, 11, 13,
17}

{13, 17, 19, 23,
29, 31}

{59, 61, 67, 71,
73, 79}

{251, 257, 263,
269, 271, 277}

5 {2, 3, 5, 7, 11,
13, 17}

{5, 7, 11, 13, 17,
19, 23}

{23, 29, 31, 37,
41, 43, 47}

{79, 83, 89, 97,
101, 103, 107}

6 {2, 3, 5, 7, 11,
13, 17, 19}

{3, 5, 7, 11, 13,
17, 19, 23}

{11, 13, 17, 19,
23, 29, 31, 37}

{31, 37, 41, 43,
47, 53, 59, 61}

Table 6.2. Results of the simulation of the signal propagation time through the circuit, ps

Number of
working
moduli

Algorithm
Working range size, bits

8 16 24 32

4
I 25097 39429 84673 119623
II 11369 15876 21509 31825
III 11476 15707 24436 29211

5
I 22903 36698 76512 111360
II 12424 15646 23134 27172
III 12374 16354 22028 27055

6
I 29077 41193 66569 66711

III 15867 17340 22252 27402
IV 15542 18717 21620 26845

Table 6.3. Results of the simulation of the used area, µm2

Number of
working
moduli

Algorithm
Working range size, bits

8 16 24 32

4
I 60759 132332 319916 538922
II 186419 393166 816822 1470043
III 148011 314708 684626 1206436

5
I 62136 131767 287343 493900
II 258575 422888 877680 1448269
III 200183 347296 715083 1329741

6
I 94223 155115 282315 399128
II 444720 566191 916218 1393492
III 343388 473257 783562 1281555

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



38 V. KUCHUKOV, M. BABENKO, S. AL-GALDA

Table 6.4. Results of the simulation of the required power, W

Number of
working
moduli

Algorithm
Working range size, bits

8 16 24 32

4
I 4,82E-01 1,24E+00 3,37E+00 6,25E+00
II 1,92E-01 8,36E-01 2,67E+00 7,03E+00
III 1,42E-01 6,12E-01 1,98E+00 5,45E+00

5
I 4,75E-01 1,29E+00 3,11E+00 5,68E+00
II 2,96E-01 8,73E-01 2,91E+00 7,26E+00
III 2,14E-01 5,94E-01 2,27E+00 6,27E+00

6
I 8,24E-01 1,57E+00 2,95E+00 4,51E+00
II 8,59E-01 1,37E+00 3,32E+00 6,95E+00
III 5,32E-01 9,80E-01 2,56E+00 5,60E+00

7. CONCLUSION

From the simulation it can be seen that the method using a single redundant modulo reduces
the area used, but the running time of the algorithm increases significantly. Using the
approximate method improves the performance significantly, but has a larger area used.

Adapting the approximate method based on CRT to the single redundant modulo method
can significantly improve the computational speed compared to the single redundant modulo
method and reduce the area used compared to the approximate method with two redundant
moduli. However, the disadvantage of this method is the limitation imposed by the redundant
modulo.

One possible application of this approach is distributed data storage systems, such as
the one described in Russian patent 2780148. In this case, in order to ensure reliability, the
residue of the redundant modulo may be stored in the user’s storage, while the residues of the
working moduli are distributed to cloud storage. Thus, in case of an error or non-receipt of
one of the file parts from the distributed environment, the original data can be restored in the
manner described above.

A direction for further research may be to develop a single redundant modulo error
correction method where the error on the control modulo can also be corrected.

ACKNOWLEDGEMENTS

This work was supported by the Russian Science Foundation 19-71-10033,
https://rscf.ru/project/19-71-10033/.

REFERENCES

1. Tolstyakov, V. S. (1972) Obnaruzhenie i ispravlenie oshibok v diskretnyh ustrojstvah
[Detection and correction of errors in discrete devices]. Moscow, USSR: Sov. radio, [in
Russian].

2. Stakhov, A. P. (1977) Vvedenie v algoritmicheskuyu teoriyu izmereniya [Introduction to
the algorithmic theory of measurement]. Moscow, USSR: Sov. radio, [in Russian].

3. Akushsky, I. Ya. & Yuditsky, D. I. (1968) Mashinnaya arifmetika v ostatochnyh klassah
[Machine arithmetic in residual classes]. Moscow, USSR: Sov. radio, [in Russian].

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



STUDY OF A RRNS FOR SINGLE ERROR CORRECTION 39

4. Diamond, J. M. (1955). Checking codes for digital computers, Proceedings of the
Institute of Radio Engineers, 43(4), 487–488.

5. Brown, D. T. (1960). Error detecting and correcting binary codes for arithmetic
operations, IRE Transactions on Electronic Computers, 3, 333–337.

6. Chang, C. H., et al. (2015) Residue number systems: A new paradigm to
datapath optimization for low-power and high-performance digital signal processing
applications, IEEE circuits and systems magazine, 15(4), 26–44.

7. Haron, N. Z. & Hamdioui, S. (2011) Redundant residue number system code for fault-
tolerant hybrid memories, ACM journal on emerging technologies in computing systems
(JETC), 7(1), 1–19.

8. Tay T. F. & Chang, C. H. (2017) Fault-tolerant computing in redundant residue number
system, Embedded systems design with special arithmetic and number systems, 65–88.

9. Aremu, I. A. & Gbolagade, K. A. (2017) Redundant residue number system based
multiple error detection and correction using Chinese remainder theorem (CRT),
Software Engineering. 5(5), 72–80.

10. Gladkov, A., et al. (2022) Modified Error Detection and Localization in the Residue
Number System, Programming and Computer Software, 48(8), 598–605.

11. Soderstrand, M., Vernia, C. & Chang, J. H. (1983) An improved residue number system
digital-to-analog converter, IEEE transactions on circuits and systems, 30(12), 903–907.

12. Cherviakov, N. I. (2011). Metody, algoritmy i tekhnicheskaya realizaciya osnovnyh
problemnyh operacij, vypolnyaemyh v sisteme ostatochnyh klassov [Methods,
algorithms and technical implementation of the main problem operations performed
in the residue number system], Infokommunikacionnye tekhnologii, 9(4), 4–12. [in
Russian]

13. Cherviakov, N. I., Babenko, M. G., Lyakhov, P. A. & Lavrinenko, I. N. (2013)
Priblizhennyj metod opredeleniya znaka chisla v sisteme ostatochnyh klassov i ego
tekhnicheskaya realizaciya [Approximate method for determining the sign of a number
in the residue number system and its technical implementation]. Nauchno-tekhnicheskie
vedomosti Sankt-Peterburgskogo gosudarstvennogo politekhnicheskogo universiteta.
Informatika. Telekommunikacii. Upravlenie. 4(176), 131–141. [in Russian]

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)


	Introduction
	REDUNDANT RESIDUE NUMBER SYSTEM
	CORRECTION OF AN ERROR WITH ONE REDUNDANT MODULO
	APPROXIMATE METHOD BASED  ON THE CHINESE REMAINDER THEOREM
	APPLICATION OF THE APPROXIMATE METHOD TO ERROR CORRECTION WITH ONE REDUNDANT MODULO
	MODELLING
	Conclusion