A concatenated coding scheme for error control in data communications is analyzed. The inner code is used for both error correction and detection, however the outer code is used only for error detection. A retransmission is requested if the outer code detects the presence of errors after the inner code decoding. The probability of undetected error of the above error control scheme is derived and upper bounded. Two specific exmaples are analyzed. In the first example, the inner code is a distance-4 shortened Hamming code with generator polynomial (X+1)(X(6)+X+1) = X(7)+X(6)+X(2)+1 and the outer code is a distance-4 shortened Hamming code with generator polynomial (X+1)X(15+X(14)+X(13)+X(12)+X(4)+X(3)+X(2)+X+1) = X(16)+X(12)+X(5)+1 which is the X.25 standard for packet-switched data network. This example is proposed for error control on NASA telecommand links. In the second example, the inner code is the same as that in the first example but the outer code is a shortened Reed-Solomon code with symbols from GF(2(8)) and generator polynomial (X+1)(X+alpha) where alpha is a primitive element in GF(z(8))

Lin, S.

English

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https://ntrs.nasa.gov/search.jsp?R=19850013691 2020-03-20T18:33:49+00:00Z
(NISI-CB-1755v-4) a CCNC1TENlTEL CODING
SCHBHE FCB BBBOB CCNTBOL (Eavali Univ.,
Manoa.)	 12 p HC 102/BF 101	 CSCL 09B Uncla s
G3/61 14486
N85-22001
A CONCATENATED CODING SCHEME FOR
ERROR CONTROL
Technical Report
to
NASA
Goddard Space Flight Center
Greenbelt, Maryland
Grant Number NAG 5-407
Shu Lin
Principal Investigator
Department of Electrical Engineering
University of Hawaii at Manoa
Honolulu, Hawaii 96822
March 20, 1985
A CONCATENATED CODING SCHEME FOR
ERROR CONTROL
ABSTRACT
In this paper, a concatenated coding scheme for error control in data
communications is analyzed. In this scheme, the inner code is used for both
error correction and detection, however the outer code is used only for error
detection. A retransmission is requested if the outer code detects the pres-
ence of errors after the inner code decoding. In this paper, the probability
of undetected error of the above error control scheme is derived and upper
bounded. Two specific example schemes are analyzed. In the first example
scheme, the inner code is a distance-4 shortened Hamming code with generator
polynomial (X+1)(X6+X+1) - X 7+X 6+X 2+1 and the outer code is a distance-4
shortened Hamming code with generator polynomial (X+1)(X15+X14+X13+X12+X4+X3
+X 2 +X+1) - X 16+X 12+X 5+1 which is the X.25 standard for packet-switched data
network. This example scheme is proposed for error control on NASA tele-
command links. In the second E ,-ample scheme, the inner code is the same as
that in the first example scheme but the outer code is a shortened Reed-
Solomon code with symbols from GF(2 8 ) and generator polynomial (X+1)(X+ a)
where a is a primitive element in GF(2 8 ). We show that both example schemes
provide very high reliability.
1. Introduction
Consider a concatenated coding scheme for error control for a binary sym-
metric channel with bit-error -rate E<1/2 as shown in Figure 1. Two linear
block codes, C  and Cb , are used. The inner code C f, called frame code, is an
(n,k) code with minimum distance d f . The frame code is designed to correct
t or fewer errors and simultaneously detect a(X>t) or fewer errors where t+X+1<
d f . The outer code C  is an (n b ,k b ) code with minimum distance d  and nb mk,
where m is a positive integ er. The outer ccde is designed for error detection
only.
The encoding is done in two stages. A message of k b bits is first encoded
into a codeword of n  bits in the outer code C b . Then the nb bi, wn:d is
divided into m k-bit segments. Each k-bit segment is encoded into an n-bit
word in the frame code C f . This n-bit word is called a frame. Thus, cor-
responding to each kb -bit message at the input of the outer code encoder, the
output of the frame code enc , der is a sequence of m frames. This sequence of
m frames is called a blcck. A two dimensional block format is depicted in
Figure 2.
The decoding consists of error correction in frames and error detection
in m decoded k-bit segments. When a frame in a block is received, it is
decoded based on the frame code C f . The n-k parity bits are then removed from
the decoded frame, the k-bit decoded segment is stored in a. buffer. If there
are t or fewer transmission errors in a received frame, the errors will be
corrected and the decoded segment is error free. If there are more than a
errors in a received frame, the decoded segment may contain undetected
errors, After m frames of a block have been decoded, the buffer contains m
k-bit decoded segments. Then error detection is performed on these m decoded
segments based on the outer code C b . If no error is detected, the m decoded
-2-
segments are assumed to be error free and are accepted by the receiver. If
the presence of errors is detected, the m decoded segments are discarded and
the receiver requests a retransmission of the rejected block. Retransmission
and decoding process continues until a transmitted block is successfully
received. Note that a successfully received block may be either error free or
contains undetectable errors.
The error control scheme described above is actually a combination of
forward-error-correction (FEC) and automatic-repeat-request (ARQ), called a
hybrid ARQ scheme [1). The retransmission strategy determines the system
throughput, it may be one of the three basic modes namely, stop-and-wait,
go-back-N or selective-repeat. The reliability is measured in terms of the
probability of undetected error after decoding.
In this paper, the probability of undetected error of the above error
control scheme is derived and upper bounded. Two specific example schemes are
analyzed. In the first example scheme, the inner code is a distance-4 short-
ened Hamming code with generator polynomial (X+1)(Xb,X+1) = X 7+X 6+X 2+1 and the
outer code is a distance-4 shortened Hamming code with generator polynomial
(X+1) (X15 +X14 +X13 +X12 +X4 +X3 +X2 +X+l ) = X 16
+X12
+X5 +1 which is the X.25 standard
for packet-switched data network. This example scheme is proposed for error
control on NASA telecommand links. In the second example scheme, the inner
code is the same as that in the first example scheme but the outer code is a
shortened Reed-Solomon code with symbols from GF(2 8 ) and generator polyno-	 r
mial (X+1)(X+a) where a is a primitive element in GF(2 8 ). We show that both
example schemes provide very high reliability.
2. Probability of Undetected Error
The probability P f (eo ,c) that a decoded frame contains a nonzero error
vector e  after decoding is given by [2,3,4],
-3-
t
t min ( t-i,n-w)
P f (eO ,E) _	
(i)(njw)Ew - i+j (1-E)n-w+i-j
i=0	 j=0
where w is the weight of e o . The right-hand side of (1) only depends on w, t
and E, we denote the right-hand side of (1) as Qt(w,E).
Recall that a codeword in the outer code C  consists of m k-bit seg-
ments. At the receiver, error detection is performed on every m decoded seg-
ments based on Cb . Let Pb (e,E) denote the probability that the decoded word
contains an undetectable error pattern a (a nonzero codeword in C b ). For a
codeword v in C	 let v (j) denote theb,	 j-th segment of v, and let w j (v) be
the weight of the codeword in frame code C f into which v (j) is encoded.
Then it follows from (1) that for an undetectable error pattern a in a block
M
Pb (e,E)
 = II Qt(wj(e),C)
j=1
Let Pua ) (E) be the probability of undetected error for the outer code Cb.
Then
P (b) (E)
eEC -{0} Pb (e, C)	 (3)
For l <* 
<
j2<...<jh^m, consider the set of codewords in C  where nonzero
bits are confined in the j l -th segment, the j 2 -th segment, ..., and the j h th
segment. This set of codewords forms a subcode of C b , call a ( j l1 j 21 ... I
j. l )-subcode of C b and den_)ted by C b (j 1 1j 2 '	 jh)' If Cb is a cyclic or
shortened cyclic code, then
1. for h=1, all ( j 1 )-subcodes of C  are equivalent;
2. for h?2, all (j l ,j 2' ... lih)-subcodes of C b with the same 32-31,33-)2'
13h-
Ih-1 are equivalent codes and are called h-segment (j2-ill
j3-32"0••rjh-jh-1) subcodes of Cb.
	Consider a (j 1' i 2 ,...,j h )-subcode of Cb .	 Let i l ,i 2 ,...,i h,r l ,r 2 , ... ,rh
be a set of integers for which 0 <iq—	 q—<k and 
0<r <n-k with 1 <q<h. Let
—	 —
(1)
(2)
-4-
3 1 0j 2 1 ... ,jh
A (i l ,r l )(i 2 ,r 2 ) ... (ih,rh) denote the number of codewords v in Cb (j 1 Ij 2 " ..'jh)
such that, for 1LcL< h, the j q-th segment v (3q) of v has weight i q and w j(v)
q
iq +rq . Then it follows from (2), (3) and the definition of
Ajl,j2, ... ,jh	
that( i l , r I His 21 r 2 ) ... (ih,rh)
MP (b) (1) = I Ql(O,t)m-h{
	
Lud 
h=1	 1<jl<j2< ... <jh<m
C A31,j2, ... ,j h	 h
I LR ( i l , r 1 ) ( i 2 , r 2 ) ... ( ih , rh )	 II Q t (iq+rq ,E) } , (4)
h	 q=1
where
IRh
 = t((il,r1),(i2'r2),...,(ih,rh))
h
and d b-<	 i <q-'b
q=1
1<i
q-	 q-	 f- q q<k, 
0<r <n-k, d <i +r '1<q<h)
-	 -	
--
If C  is a cyclic or shortened cyclic code, then Eq. (4) can be simplified as
follows:
m	
C
	
u)	 m-h{(E) _	 Qt (0, E )
	
Pd 	 /
	h=1	 1<jl<j2< ... <j <m
9-
	
C	
A 3 1 r3 2 ,
 .... 
3h	
h
q ^^
	
I LR	 (i l ,r l H i2,r2)...(ih,rh) 0-1 
Qt (i +r E)
	 (5)
h
From (4) we see that, if we know the detail weight structure of Cb(J1,J2^
.... j h), the error probability P (b) ( E ) can be computed. However, for a
ud
	given C, it is not easy to find 
A 1132 )1	 13h2)... (ih,rh)	 To overcome thisb
difficulty, we have derived upper bounds on the terms on the right-hand side
of (5) (5). We assume that e < (t+2)/(3t+4). suppose that t=1 and the inner
code is an even-weight code and the outer code is a cyclic or shortened cyclic
even-weight code. Let {A(b)I be the weight distribution of the outer code Cb'
We have obtained the following bound on Pud(E):
-5-
... ^_„^^.....r...^......^..^_...,..s.,..^ _-..^,•^,..--_- - - ^_
	
:^r.^_.-	 ^.--^—	 - -...^ mss..
	10	 n-k
	
Pua)( E) ` m L	
CGA(i,r)Q1(i+r,E)
i=db r=0
	
m	 2
	
+ I	 (m-j+l)	 Al'3	 TI Q1(E(ip),E)
	
j=2	 ii,i2<10 1 2 p=1
1<il,i2
	
+ { 1^ 	 (A (b) -MAl) -	 (m - j+l)	 A"' )Q (4,F-) 3
i=db i	 i	 j=2	 il,i2<10 i l ,i 2 1
i<i1,i2
+ min{( 3 ) 0) 2 (3) , A (b) ^Q 1 (4,E) 3 + A(b)Q1(6,E)
+	 A4b)Q1(4,E)1 + I A4b+2Q1(4,E)i-1Q1(6,E)
	
i=4	 i=3
+ (26/nb )
-26 (1-26/nb )
nb-26
 Q l (4,E) 5Q 1 (6,E)	 (6)
where
31 r3A	 - n--k n--k	 nC-k Ajl,j2,---,jh	
( )(i ,r )(i , 2
	 h
r )...(i ,r )
h	 r 1 =0 r 2=0	 rh=0	 1 1	 2	 h
and
d f , for i < d 
i, for even i and i > d 
' i+l, otherwise.
On the other hand, it follows from (5) that
10 n-k
Pua ) (E) > m Ql(O,E)m-1 I	 I A(i,r)Ql(i+r,E)	 (9)
i=db r=0
3. Examples
We cons;der two examples of the concatenated coding scheme.
Example 1: The frame code C  is a distance-4 Hamming code with generator
polynomial,
gf M = (X+1)(X6+X+1) = X 7 +X 6+X 2+1 ,
where X 6 +X+1 is a primitive polynomial of degree 6. The maximum length of
this code is 63. This code is used for single error correction. The code is
-6-
capable of detecting all the error patterns of double and odd number errors.
The outer code is also a distance-4 shortened Hamming code with generator
polynomial,
15 14 13 12 4 3 2	 16 12 5
9 0 (X) - (X+1)X +X +X +X +X +X +X +X +1) - X +X +X +1
where X 15+X 14+X 13+X 12+X 4+X 3+X 2+X+1 is a primitive poly ► -ial of degree 15. We
assume that the number of frames in a block is greater than 3 and less than
65. The 16 parity bits of this code is used for error detection only. This
scheme is proposF,d for NASA telecommand system. For this example upper bounds
on the probabili t y of undetected error have been computed in [5j.
Example 2: This example is a variation of example 1. The frame code C  is
the same as example 1. The outer code is a shortened Reed-Solomon code with
generator polynomial (X+1)(X+a) over GF(2 8 ), where a is a root of
X8+X4+X3+X2+1.
For various E, k, and m, the bound or Dud)(E) given by (6) is evaluated
and plotted in Figures 3 to 5.
ACKNOWLEDGMENT
The authors would like to thank Mr. Toyoo Takata for his helpful support.
-7-
REFERENCES
1. S. Lin and D.J. Costello, Jr., Error Control Coding: Fundamentals and
Applications, Prentice-Hall, New Jersey, 1983.
2. E.R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968.
3. F.J. MacWilliams, 'A Theo-em on the Oistribution of Weights in Systematic
Code," Bell System Technical Journal, Vol. 42, pp. 79-94, 1963.
4. Z. McHuntoon and A.M. Michelson: "On the Computation of the Probability
of Post-Decoding Error Events fo: Block Codes,' IEEE Trans. on Informa-
tion Theo, Vol. IT-23, No. 3, May 1977, pp. 399-403.
5. A. Kitai, 'A Method for Computing Probability of Undetectable Error of
Error Correcting Code," Thesis for M.E. Degree, Dept. of Information and
Computer Sciences, Osaka University, 1984.
6. D.J. Costello, Jr., private communication.
-8-
Figure 1 A concatenated coding scheme
n— k
M
Segment	 111^\Rll
INNON'
Block Parity-Check Sits
. 7/Z-/ M/1, /
. — Frame
Frame Parity-
Check Bits
Figure 2 Block format
Upper bounds on the pr pbab111ty
of undetected error Pud)(c)
Inp^	
Y-1 (RS)
T-5 (RS)
110	 ,
6
16 4 1 Y-7 (HM)	 / Y
-3 (RS).!
IOK	
Y-5 (HM)
Ion Y -3 (HH)
10 a	 Number of frames per block
10	 -20	 30	 4o	 "So	 60
Figure 3 Upper bounds on the probability
of undetected error for bit
error rate = - 10-4.
Y: the number of information bytes in
a frame
HM: example 1	 RS: example 2
Upper bounds on the pr pbabillty
of undetected error Pub)(c)
16,
Y-7 (RS)
10	
l
to	 s
Y-5 (RS) Y 1_•-^-
16 	
Y-5 (HM)
lo" Y-3 (RS)
Y-3 (HM)
to "
Number of frames per block
to	 ]0	 30	 40	 5o	 6U
Figure 4 Upper bounds on the probability
of undetected error for bit
error rate E = 10-5.
Y: the number of infcryration bytes in
a frame
HM: example 1
	 RS: example 2
Upper bounds on the prppbab111ty
of undetected error Pua)(c)
l ''o 	
T-7 (P")
b	 "-7 (HM)
r
(RS)
low
•l Y-S 
lo"
ki *
10 )I
w'°
1)	 :+0	 30	 LW)	 SO	 to
Figure 5 Upper bounds on the probability of undetected error
rate E = 10-6.
Y: the number of information bytes in a frame
HM: example 1
	
RS: example 2
Y-5 (HM)
Y-3 (RS)
r
T • 3 (HM)
Number of frames per block


A concatenated coding scheme for error control

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