CHAPTER-4
A BROAD CLASS OF ADDITIVE ERROR CODING,
CHANNELS AND LOWER BOUND ON THE PROBABILITY
OF ERROR FOR BLOCK CODES USING SK- METRIC
The contents of this Chapter are based on my following published paper:
Gaur A., Sharma B.D., “A Broad Class of Additive Error Coding, Channels and
Lower Bound on the Probability of Error for Block Codes using SK- Metric”,
International Journal of Applied Mathematics and Statistics, vol. 52, issue-1,
pp. 119-131, 2014.
54
CHAPTER-4
A BROAD CLASS OF ADDITIVE ERROR CODING,
CHANNELS AND LOWER BOUND ON THE PROBABILITY
OF ERROR FOR BLOCK CODES USING SK- METRIC
4.1 INTRODUCTION
In previous chapters we defined a new way of looking for error that actually occurs
during the transmission and obtained bounds on length and parity checks for newly defined
error patterns. In practice this calls for visiting the channels which match with these errors or
move generally with SK-partition of GF (q ) . We deal with this problem in this chapter.
Hamming metric based studies, it may be recalled that Hamming metric was initially
proposed for binary codes and thereby for binary channels, and that its use was
straightforwardly extended to non-binary cases, without exploring other possible metrics for
q-nary cases. Another metric, the Lee-metric was, of course, also introduced in a rather adhoc manner. But this basic tool, the metric employed, of measuring possible errors continued
without deeper study. As mentioned earlier, this gap was filled by Sharma and Kaushik by
exploring all possible metrics for q-nary cases, q > 2 . They introduced a class of distances,
that includes Hamming and Lee distances as particular cases. With choices available, it is now
possible to formulate errors in terms of an SK-distance that best fit the channel, and thus
revisits the efficiency issues and improve the bounds, sorting out only those (partial to those
under Hamming metric) errors that actually happen digit-wise rather than all that that would
happen if Hamming distance was considered. In 1979 Kaushik [61] obtained Hamming,
Plotking and Varshamov-Gilbert type of bounds using SK-distance criteria, in Chapter-3, we
have derived a combinatorial upper bound on number of parity check digits for linear codes
that correct all what we call ‘partial random error’ of an (n, k ) code with minimum SKdistance at least d . Our result generalized illustrious Varshamov-Gilbert bound, which
follows from it as a particular case.
In this Chapter using SK-metric approach we generalize the idea of additive errors by
introducing a broad class of ‘Class-additive errors’. In the setting of SK-distances and error
55
probabilities, we generalize the concept of ‘binary-symmetric-channel’ to what is ‘q-nary-SKPartition Symmetric Channel’. We study the probabilistic aspects of error controlling codes
and report some results on bounds on probability of error for block codes developed on the
q-nary SK- Partition Channel.
4.2 A GENERALIZATION OF BINARY SYMMETRIC CHANNEL
Since SK-based studies consider partitions of Z q for distance etc., a Z q channel
corresponding to an SK partition℘ = {B0 , B1 ,..., Bm −1} of Z q , q prime, the probabilities p( j / i )
have to be related to the partition℘ , which we define as follows:
DEFINITION: SK-DISCRETE MEMORYLESS CHANNEL
Given an SK-partition℘ = {B0 , B1 ,..., Bm −1 } of Z q , q prime, the SK-discrete memoryless
channel has the transition probabilities given by
p( j (received ) | i (sent )) =
p ( j − i ) p (Bl )
=
Bl
Bl
for i, j ∈ Z q , j − i ∈ Bl ,0 ≤ l ≤ m − 1, mod q ,
where p ( Bl ) is probability of sub-set Bl , 0 ≤ l ≤ m − 1 , in the partition℘ = {B0 , B1,..., Bm −1} of Z q .
We shall denote this q-nary symmetric channel for SK-partition ℘ by ‘q℘SC’ and it will
have channel matrix given by,


p ( Bl )
: j − i = l , .
C q℘ =  p ( j / i ) = p ( j − i ) / i =
| Bl |

 q× q
NOTE: It may be noted that in the definition adopted above the error probability is associated
with the class to which error (given by i − j mod q) really belongs. Also it may be quickly
seen that for binary case and Hamming distance, the channel reduces to Binary-SymmetricChannel. In this sense the definition above generalizes the BSC for q − nary case depending
56
upon the underlying SK-partition. That this is also symmetric in nature may be easily verified,
and is illustrated by an example given below.
EXAMPLE 4.1:
Let Z 7 = {0,1,2,3,4,5,6}, and the SK-partitions of Z 7 , be given by,
℘ = {B0 , B1 , B2 } ,
where,
B0 = {0} , B1 = {1,2,5,6} and B2 = {3,4} ,
so that B1 = 4 and B2 = 2 then with B0 , B1 , B 2 having respectively the probabilities p 0 ,
p1 , p 2 , so that p0 + p1 + p2 = 1 , the channel matrix is given by
j
i
0
1
2
5
6
3
4
0
1
2
5
6
3
4
p0
p1
B1
p1
B1
p1
B1
p1
B1
p2
B2
p2
B2
p1
B1
p1
B1
p2
B2
p1
B1
p2
B2
p1
B1
p1
B1
p1
B1
p2
B2
p2
B2
p2
B2
p1
B1
p1
B1
p1
B1
p1
B1
p1
B1
p1
B1
p1
B1
p1
B1
p1
B1
p2
B2
p2
B2
p0
p1
B1
p2
B2
p1
B1
p1
B1
p2
B2
p0
p2
B2
p2
B2
p1
B1
p1
B1
p0
p1
B1
p1
B1
p1
B1
p0
p2
B2
p1
B1
FIGURE 4.1: Channel Matrix
57
p0
p1
B1
This channel, in terms of notation adopted is a ‘7℘ SC’, where,
q=7
and
℘ = {(0), (1,2,5,6), (3,4)} .
4.3 CODES WITH SK-PARTITION CLASS ADDITIVE ERRORS
In this section, we define errors arising from addition of specific classes of an SKpartition. This approach generalizes several ideas including what we call ‘Broadly
Generalized SK-distance Perfect Codes’ studied in chapter-5.
B i - ADDITIVE ERROR CODE
With the introduction of SK-partition, it is possible to consider various different types of
errors. The errors being additive in nature, as these are, for example, there may be errors that
arise by addition of digits occurring in a particular partitioning set Bi .If the errors considered
are in all positions limited to addition of elements of Bi , we shall call such a code as ‘ Bi Additive-Error Code’.
DEFINITION: B i - ADDITIVE ERROR
Given a vector u = (a1 , a 2 ,..., a n ) over Z q , and an SK-partition℘ = (B0 , B1 ,..., Bm −1 ) , the vector
u will be called to have Bi -additive errors, if the positions in vector u undergo addition of
digits in the subset Bi .
EXAMPLE 4.2: Suppose we have Z 13 = {0,1,2,...,12} , with q = 13 . Let us consider the SKpartition of Z 13 given by
℘ = {B0 , B1 , B2 , B3 } ,
where
B0 = {0} B1 = {1,2,11,12} , B2 = {3,4,9,10} and B3 = {5,6,7,8},
Let uˆ = (0,2,5,4,7 ) , then û has B1 additive error in positions 3rd and 4th, and û becomes
(0,2,5 + 1,4 + 4,7 ) or (0,2,5 + 11,4 + 2,7 ) ,
58
where all addition are mod 13.
In general this idea of additive-errors may be extended further to addition of digits of
more than one partitioning set. So there can be:
B1 -Additive errors, or more specifically B1 -additive single-errors, B1 -additive double-errors,
or more generally, depending on the number of positions in which errors are limited specified.
B1 ∪ B 2 -additive errors if the additive errors are digits from union of B1 and B2 , etc.
NOTE: When the errors are not limited to specific partitioning set or sets, this, as may be
clearly visualized, will correspond to what will naturally be the case leading to Hammingmetric.
BROADLY GENERALIZED SK-DISTANCE PERFECT CODES
With SK-metric, in Chapter-5 we have studied, what may in the present context be called:
a. B1 -additive e-error perfect codes, that is those codes which correct e or less B1 additive errors and no more.
b. B1 ∪ B 2 -additive error perfect codes, that is those codes which correct e or less B1
and B2 errors and no more.
CLASS-QUASI PERFECT CODES
It may be recalled that in classical studies of codes, a quasi-perfect code is defined as follows:
DEFINITION: QUASI PERFECT e-ERROR CORRECTING CODE
A quasi perfect e-error correcting code corrects all errors in e or fewer positions, some in
e + 1 positions and none in more than e + 1 positions.
With errors considered in terms of classes of an SK-partition and consequent metric, a
quasi perfect e-error correcting code can be defined in several ways.
Corresponding to a B1 -additive e-error, a quasi perfect codes e-error B1 -additive can be
considered in following different cases:
59
(
)
DEFINITION: B1 -ADDITIVE e B1 , e + 1B1 CLASS QUASI PERFECT CODE
A code that corrects all B1 -additive e or less errors and the remaining only B1 -additive in
e + 1 positions is a B1 -additive (e B , e + 1B ) class quasi perfect code. These codes, if they
1
1
exist for given n, q, k, e, and the partition, will satisfy the condition
 n
n
n
2
q n − k − 1 +   B1 +   B1 + ... +   B1
e 
2
 1 
(
e
 n 
<
  e + 1 B1

 
e +1
(4.1)
)
eB1 CLASS QUASI PERFECT CODE
DEFINITION: B1 -ADDITIVE e B1 , ~
A code that corrects all B1 -additive e or less errors and the remaining ones are all in e or less
(
)
eB1 class quasi
positions with entries from other than elements of B1 is a B1 -additive e B1 , ~
perfect code. The code gives a combined set of S digits:
 n
n
n
2
q n − k − 1 +   B1 +   B1 + ... +   B1
e 
2
 1 
(i)
e
  n
n
n
 <    S +   S 2 + ... +   S
 
2
e 
  1 
e

 (4.2)


One can consider following further sub-cases of (4.2):
If all the others are from B2 then S = B2 and then the left-hand expression in
(4.2) be the following:
 n
n
n
2
   B2 +   B2 + ... +   B2
 1
2
e 
 
(ii)
e

.


(4.3)
Now in case (b) if all the others are any entry other than from B1 then
S = q − 1 − B1 and then the left-hand expression in (4.2) is the following:
 n
n
n
2
   (q − 1 − B1 ) +   q − 1 − B1 + ... +   q − 1 − B1
 1
2
e 
 
e

.


(4.4)
(iii) Yet differently, if all the others are such that at least one is from B2 , while
others are from B1 and then the left-hand expression in (4.2) be the following:
60
 n
n
   B2 +   B2
 1
2
 
(
B1 + B2
) + ... +  n 
e 
B2
(
B1 + B2
)
e −1

.


(4.5)
From the above discussion, it can be seen that in this approach there can be very many
different classes of the quasi-perfect codes arising from an underlying SK-partition of Z q .
Before coming to a bound on probability, we recall two theorems given below proved by us in
chapter 2, for errors considered above.
THEOREM A: Given an SK-partition ℘ = {B0 , B1 ,..., Bm −1 } of Z q , q prime, a necessary
condition for existence of an (n, k) code over Z q , correcting errors in e-positions each of
weight one is given by
t
 e n 
n
n−k
n
−
≥
log
k


or
≤
B
q
q ∑ 
∑
 t  B1  .
  1
t
=0
t =0  t 
   
e
t
THEOREM B: Given an SK-partition ℘ = {B0 , B1 ,..., Bm −1 } of Z q , q prime, a necessary
condition for the existence of an (n, k) code over Z q , correcting error in e or less positions so
that in these positions an entry j when in error is received as an element of B1 + j or B2 + j
is given by
n
∑  e  (
e
t =0
 
B1 + B2
)
e
≤ q n−k .
4.4 LOWER BOUND ON THE PROBABILITY OF ERROR FOR SKMETRIC CHANNEL
In this section we take up studies of error-probabilities under additive SK-Partition Class
errors and related bounds.
61
TWO CATEGORIES OF ERROR-PATTERNS
Considering communication over the SK-Symmetric channel defined above, we
consider errors in the following two categories and their probabilities:
i.
R1 = Number of non-correctable error patterns with errors in e or less positions
with entries arising by addition of digits other than those of B1 .
ii.
R2 = Number of non-correctable error patterns with errors in e + 1 or more
positions arising by addition of entries from any of Bi ’s.
iii.
P(R1 ) = Probability of non-correctable error patterns with errors in e or less
positions with entries arising by addition of digits other than those of B1 .
iv.
P(R2 ) = Probability of non-correctable error patterns with errors in e + 1 or more
positions arising by addition of entries from any of Bi ’s.
THEOREM 4.1: Given an SK-partition ℘ = {B0 , B1 ,..., Bm −1 } of Z q , q prime, probability of
error in transmitting over ‘q℘ SC’, a code capable of correcting e B1 -additive errors, on
receiving vectors that have errors in e or fewer positions with addition in those positions of
digits other than those of B1 is
 x 


x
y
−


( (x + p )
0
n ,e
+ ( y + po )
n ,e
),
where k = q − 1 − B1 , q 01 = 1 − p 0 − p1 , q1 = 1 − p1 , kq01 = x , B1 q1 = y .
PROOF: We begin by counting the error-patters of the category R1 , that are non-zero and
non- B1 additive in e or less positions, and then consider corresponding probability of their
being received over the channel.
Considering an e error vector with one non-zero entry other than those for B1 will have n − 1
entry for q − 1 − B1 .
Let E1 denote the number of n-vectors that have one non-zero entry other than those from B1 ,
then
62
n
E1 =   ( q − 1 − B1
1 
).
Next, the probability of receiving these E1 vectors is
n
P1 =   k q 01 p 0n −1 ,
1 
where p 0 is the probability of correct transmission of a digit and q 01 is the probability of
change other than addition of that of from B1 i.e. q 01 = 1 − p 0 − p1 .
Similarly, we can find E 2 , E 3 …. E e and corresponding probabilities of their receiving. For
these errors in r positions, in general we have
(
)
 n
r −1
Pr =   k r q01r + k r −1 B1 q 01r −1 q1 + k r − 2 B1 2 q 01r − 2 q12 + ... + k B1 q 01 q1r −1 p 0n − r
r 
We can, with a view to obtain close expression, suppose kq01 = x and B1 q1 = y then,
2
r −1
n   y   y 
 y 
Pr =   x r 1 +   +   + ... +    p 0n − r
 x  
 r    x   x 
  y r
1−  
n   x 
=   
 r   1 − y 
 
x



 n−r  n   x x r − y r
 p 0 =  r   ( x − y )
 



(
)  p


n−r
0
.
So the total probability of receiving vectors of this category in error, then is
e
∑P
r =1
r
 x 

= 
x− y
 e  n  r n−r e  n  r n−r
 ∑   x p 0 − ∑   y p 0

r =1  r 
 r =1  r 
 x 

= 
x− y
( (x + p )
0
63
n ,e
)
+ ( y + p o ) n ,e .




Where, (a + b ) n ,e stands for e + 1 (e < n ) terms in the binomial expansion (a + b ) n.
THEOREM 4.2 : Given an SK-partition ℘ = {B0 , B1 ,..., Bm −1 } of Z q , q prime, probability of
error in transmitting over ‘q℘ SC’, a code capable of correcting r ≥ e + 1 B1 -additive errors,
on receiving vectors with addition in those positions of digits from any of Bi ’s is
r
 n   m −1

   ∑ Bi qi  p 0n − r , r = e + 1, e + 2,..., n ,
∑
r = e +1  r   i =1

n
where, qi = 1 − pi , i = 1,2,..., m − 1 .
PROOF: We start with finding expressions for probability of receiving vectors with non-zero
r entries. This for r = e + 1 is given by the following expression,
 n 

Pe +1 = 
 e + 1
(
B1 q1 + B2 q 2 + ... + Bm −1 q m −1 ) e +1 p 0n −(e +1) .
For a general value of r, this can be put as,
r
 n   m −1

Pr =    ∑ Bi qi  p 0n − r , r = e + 1, e + 2,..., n .

 r   i =1
The total probability of receiving vectors of this category for r ≥ e + 1 , we have the probability
of error given by,
r
 n   m −1
 n−r

P
=
∑
∑
r
 r   ∑ Bi qi  p 0 .
r = e +1
r = e +1    i =1

n
n
COROLLARY 4.1: Given an SK-partition ℘ = {B0 , B1 ,..., Bm −1 } of Z q , q prime, probability
of error in transmitting over ‘q℘ SC’, a code capable of correcting r B1 -additive errors, on
receiving vectors that have errors in e + 1 or more positions with addition in those positions of
digits from any non zero entry at least
64
n
n
∑  r  (q − 1)
r = e +1
 
r
q 0r p 0n − r .
PROOF: In general, we may assume that p 0 > pi , i =1,2,…, m-1 then 1 − pi = qi > q 0 , then
n
Pr =   ( B1 q1 + B2 q 2 + ... + Bm −1 q m −1
r 
)
r
p 0n − r
n
≥   ( B1 q 0 + B2 q 0 + ... + Bm −1 q 0 ) r p 0n − r
r 
n
=   (q − 1) r q 0r p 0n − r .
r 
The total probability of receiving vectors of this category for r ≥ e + 1 , we have the probability
of error given by,
n
∑ Pr =
r = e +1
n
n
∑  r  (q − 1) q
r = e +11
r
 
r
0
p 0n − r .
THEOREM 4.3: Given an SK-partition ℘ = {B0 , B1 ,..., Bm −1 } of Z q , q prime, for a code
capable of correcting B1 -additive errors in e positions, probability of error perror , in
transmitting over ‘q℘ SC’ is,
p error
where
 x 

≥ 
x− y



 n 

 ( x + p 0 ) n ,e + ( y + p o ) n ,e + 
 B1 − α e +1  q1e +1 p 0n −(e +1) 



 e + 1


,
n
n


r r n−r
 + ∑  r  (q − 1) q 0 p 0

 r =e + 2  

(
)
n
n
n

2
α e+1 = q n− k − 1 +   B1 +   B2 + ... +   B1 e  .
2
e 

 1 
65
PROOF: Earlier, for the code under consideration, we have considered two types of errorpatterns so the total number of error patterns is R = R1 + R2 with their corresponding
probability of error
 x 

Pr + ∑ Pr = 
∑
r =1
r = e +1
x− y
e
n
( (x + p )
n ,e
0
+ ( y + po )
n ,e
)
r
 n   m −1

+ ∑    ∑ Bi qi  p 0n − r .
r = e +1  r   i =1

n
For obtaining a bound on the probability of error we consider that the code is quasi perfect
that is it is a code capable of correcting B1 -additive errors in e or fewer positions and others
entries are all B1 -additive in e + 1 positions and no more. The number of correctable error
patterns up to e positions is
 n
n
n
2
1 +   B1 +   B1 + ... +   B1

2
e 
 1 
e

.


And the number of the remaining correctable error-vectors is,
 n
n
n
2
α e+1 = q n − k − 1 +   B1 +   B1 + ... +   B1
2
e 
 1 
e

.


Also the number of all other non-correctable error patterns, that is those outside the quasi
perfect consideration, are some of errors in e + 1 positions and the rest those having any of the
q − 1 , non-zero entries in e + 2 or more positions.
The number of those having non-zero entries in e + 2 and more positions are:
n
n
∑  r  (q − 1)
r =e + 2
 
r
q 0r p 0n − r .
It may thus be seen that R2 has error vectors that non-correctable some of which are of e + 1
errors and all others of non-zero e + 2 or more positions.
For the purposes of getting a lower bound, the probability of error of receiving R2 vectors
correcting B1 -additive errors in transmitting over ‘q℘ SC’ is
66
n
 n 

n
 B1 − α e +1  q1e +1 p 0n −(e +1) + ∑   (q − 1) r q 0r p 0n − r .
p error ( R2 ) ≥ 
r =e + 2  r 
 e + 1

Considering different classes of error patterns together, the probability of error perror in the
code under consideration has the lower bound given by
 x 

 

 (x + p 0 ) n ,e + ( y + p o ) n ,e
 x − y 

≥
.
n
n
 +  n  B − α  q e +1 p n −(e +1) +
r r
n−r 
∑   (q − 1) q0 p0 
e +1  1
0
  e + 1 1
r =e + 2  r 


 

(
p error
)
This proves the result.
HAMMING CASE
Above considerations when applied to Hamming case compare [78], mean that the class R1
will be empty, so that
 x 


x− y
( (x + p )
n ,e
0
)
+ ( y + p o ) n ,e = 0.
And now the bound takes the form:

p error ≥ 


n

 n 

n
 B1 − α e +1  q1e +1 p 0n −(e +1) + ∑  (q − 1) r q 0r p 0n − r  .
 

r =e + 2  r 
  e + 1


The result proved above is specific for B1 additive errors and the quasi-perfect consideration
taken there. It can be extended to other class additive errors and situations. In the next
theorem, we consider when it is B1 ∪ B2 additive.
THEOREM 4.4: Given an SK-partition ℘ = {B0 , B1 ,..., Bm −1 } of Z q , q prime, probability of
error in transmitting over ‘q℘ SC’, a code capable of correcting e B1 ∪ B2 -additive errors, on
receiving vectors that have errors in e or fewer positions with addition in those positions of
digits other than those of B1 and B2 is
67
 x1   e  n  i
  ∑   x1 − y1i p 0n −i


 x1 − y1   i =1  i 
(
Where k1q012 = x1 and
(
B1 + B2
) (q
1
)

,


+ q 2 ) = y1 , k1 = q − 1 − B1 − B2 ,
q012 = 1 − p0 − p1 − p 2 , q1 = 1 − p1 , q 2 = 1 − p 2 .
PROOF: We begin by counting the errors patterns and then corresponding probability of
their receiving over the channel.
Considering an error vector with one non-zero entry other than those from B1 and B2 will have
n-1 non zero entry form (q − 1 − B1 − B2 ) . Their number E1 is given by
n
  (q − 1 − B1 − B2 ) .
1 
Next the probability of receiving these E1 vectors is
n
P1 =   k1 q 012 p 0n −1 .
1 
Similarly we can find E2 , E3 ,…, E e and corresponding probabilities. For these errors in r
positions in general we have
(
2
r r
r −1
r −1
r −2
B1 + B2
 n   k1 q 012 + k1 ( B1 + B2 ) q 012 (q1 + q 2 ) + k 1
Pr =  
r −1
 r   + ... + k1 B1 q 012 q1r −1 + q 2r −1
(
Suppose k1 q 012 q 012 = x1 and
)
(B
1
+ B2
) (q
1
+ q 2 ) = y1 then,
2
r −1
 y1   n − r
 n  r   y1   y1 
p0
=   x1 1 +   +   + ... +  
 r    x1   x1 
 x1  
68
2
)q ( q
r −2
012
2
1
)
+ q 22  n − r
p
 0

  y r
1−  1 
 n   x1 
=  
y1 
 r  

1
−
 
x
1 
 


 n − r  n  x1 x1r − y1r
 p 0 =  
 r  (x1 − y1 )



(
)  p


n−r
0
 x1   e  n  r n − r e  n  r n − r

  ∑   x1 p 0 − ∑   y1 p 0
p
=
∑
r

r =1
r =1  r 
 x1 − y1   r =1  r 
e
 x1 

= 
 x1 − y1 
( (x
1




)
+ p 0 ) n ,e + ( y1 + p o ) n ,e .
THEOREM 4.5: Given an SK-partition ℘ = {B0 , B1 ,..., Bm −1 } of Z q , q prime, for a code
capable of correcting (B1 ∪ B2 ) -additive errors in e positions, probability of error perror in
transmitting over ‘q℘ SC’ is

  x1 
n ,e
n ,e



(
)
x
p
y
p
(
)
+
+
+
+
o
1
0
1

  x1 − y1 
≥

n
 e +1 n −(e +1)
n
  n 
r r n−r 
+ ∑   (q − 1) q 0 p 0 
  e + 1 ( B1 ∪ B2 ) − α e +1  q1 p 0
r =e + 2  r 



 
(
p error

n
)
(
n
)
(
n−k
where α e +1 = q − 1 +   B1 ∪ B2 +   B1 ∪ B2
2
 1 
)
2
n
+ ... +   ( B1 ∪ B2
e 
)
PROOF: From theorem 4.2 and 4.4, we have,
 x 
 (x + p 0 ) n ,e + ( y + p o ) n ,e
P(R1 ) = 
x− y
(
 x1 

Pr + ∑ Pr = = 
∑
x
y
−
r =1
r = e +1
1 
 1
e
n
( (x
1
)
r
 n   m −1
 n−r
(
)
P
R
and
2 ∑ 
 r   ∑ Bi qi  p 0 then
r = e +1    i =1

n
+ p 0 ) + ( y1 + p o )
n ,e
n ,e
)
The number of correctable error patterns up to e positions is
69
r
 n   m −1

+ ∑    ∑ Bi qi  p 0n − r .
r = e +1  r   i =1

n
e

.


 n
n
1 +   ( B1 ∪ B2 ) +   ( B1 ∪ B2

2
 1 
)
2
n
+ ... +   ( B1 ∪ B2
e 
)
e

.


Number of the remaining other correctable errors is
 n
n
α e +1 = q n − k − 1 +   ( B1 ∪ B2 ) +   ( B1 ∪ B2
2
 1 
)
2
n
+ ... +   ( B1 ∪ B2
e 
)
e

.


Also the number of rest of all patterns not falling for the quasi perfect case and are in e + 2 or
higher positions is
n
n
∑  r  (q − 1)
r =e + 2
 n 

p error ( R2 ) ≥ 
1
e
+



(B
1
∪ B2
 
)−α
e +1
r
q 0r p 0n − r .
n
 e +1 n −(e +1)
n
+ ∑   (q − 1) r q 0r p 0n − r .
 q1 p 0
r =e + 2  r 

Then probability of error for a code capable of correcting
(B
1
∪ B2 ) -additive errors in
transmitting over ‘q℘ SC’ is
  x1 
 n 
 e +1 n −(e +1)
n ,e
n ,e




(
)
(
)
(
)
α
x
p
y
p
B
B
+
+
+
+
∪
−

 q1 p 0
1
0
1
1
2
1
o
e
+
 e + 1
  x1 − y1 




≥
n
n

r r n−r
 ∑  r  (q − 1) q 0 p 0
 r =e + 2  
(
p error
)

+

.



4.5 CONCLUDING REMARK
Most algebraic codes have structured to correct specific types of errors. But the channel being
probabilistic in nature, there is an area of reliable of the communication.
In this chapter we have introduced a very generalized concept of additive errors in terms
of classes of SK-partitions of Z q . From this has followed the concept of ‘Class-Additive
errors’ and ‘Broadly Generalized SK-distance Perfect Codes’. In this approach error patterns
have also been defined in various different ways and so the studies that follow from it.
70
Coding is required for different types of channels. Our study has extended the widely
used concept of Binary-Symmetric Channels. Considering two types of error-patterns, there
are results on their probabilities of errors for the q-nary channels arising from SK-partitions.
Then there are bounds on probability of errors which generalizes approach and results on
probability of errors for a code designed for different classes of errors.
With generalization of additive errors and of symmetric channel for the new setting
under SK-studies, there are several other possibilities that arise for further studies.
71