Consider an (n,k) linear code with symbols from GF(2 sup M). If each code symbol is represented by a m-tuple over GF(2) using certain basis for GF(2 sup M), a binary (nm,km) linear code is obtained. The weight distribution of a binary linear code obtained in this manner is investigated. Weight enumerators for binary linear codes obtained from Reed-Solomon codes over GF(2 sup M) generated by polynomials, (X-alpha), (X-l)(X-alpha), (X-alpha)(X-alpha squared) and (X-l)(X-alpha)(X-alpha squared) and their extended codes are presented, where alpha is a primitive element of GF(2 sup M). Binary codes derived from Reed-Solomon codes are often used for correcting multiple bursts of errors

Lin, S.

English

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https://ntrs.nasa.gov/search.jsp?R=19850010319 2020-03-20T20:09:59+00:00Z
MEN
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ON THE BINARY WEIGHT DISTRIBUTION OF
SOME REED-SOLOMON CODES
I
ti
I
I
r.
r
r
Technical Report
to
NASA
Goddard Space Flight Center
Greenbelt, Maryland
r'	 1
,1
Grant Number NAG 5-407
;I
. I
Shu Lin	 M..
Principal Investigator
	 r.
Department of Electrical Engineering
University of Hawaii at Manoa
Honolulu, Hawaii 96822
`	 February 22, 1985
I
ON THE BINARY WEIGHT DISTRIBUTION OF
SOME REED-SOLOMON CODES
Tadao Fasami	 Shu Lin
Osaka University	 University of Hawaii at Manoa
Toyonaka, Osaka 560, Japan 	 Honolulu, Hawaii 96822, U.S.A.
ABSTRACT
Consider an (n,k) linear code with symbols from GF(2 m ). If each code
symbol is represented by a m-tuple over GF(2) using certain basis for
GF(2 m ), we obtain a binary (nm,km) linear code. In this paper, we investi-
gate the weight distribution of a binary linear code obtained in this manner.
Weight enumerators for binary linear codes obtained from Reed-Solomon codes
over GF(2m ) generated by polynomials, (X-a), (X-1)(X-a), (X-a)(X-a2) and
(X-1)(X-0L)(X-n '4)
 and their extended codes arc presented, where a is a
primitive element of GF(2m ). Binary codes de-ived from Reed-Solomon codes
a:e often used for correcting multiple bursts of errors.
1
r	 ^
1. Introduction
Let {B1'B2'...,Bm} be a basis of the Galois field GF(2 m ). Then
each element z in GF(2 m ) can be expressed as a linear sum of R1,B2,...,Bm as
follows:
Z - c1B1 + c2B2 + ... + cmBm,
where c 
1.EGF(2) for 1<i<m. There is a one-to-one correspondence b y ween the- -
element z and the m-tuple (c 1,c2,...,cm) over GF(2). Thus z can b repre-
sented by the m-tuple (c l ,c 2" "' cm ) over GF(2).
Let C be an (n,k) linear block code with symbols from the Galois field
GF(2 m ). If each code symbol of C is represented by a m-tuple over the
binary field GF(2) using the basis {B1162,...,Bm} for GF(2 m ), we obtain a
binary (mn,mk) linear block code C b . If code C is capable of correcting t
or fewer random symbol errors, then C  is capable of correcting any
combination of
X -	 r_
1+((Q+m-2)/mi
or fewer bursts of errors of length Z [1].
In this paper, we investigate the weight distributions of binary codes
deriv I from codes with symbols from GF(2m ). Weight enumerators for binary
codes obtained from Reed-Solomon codes over GF(2 m ) generated by polynomials,
(X-a), (X-1)(X-0), (X-0)(X-a2) and (X-l)(X-a)(X- a2 ) and their extended
codes are presented, where a is a primitive element of GF(2m).
2. Binary Weight Distributions of Linear Block Codes over GF(2m)
Let C be an (n,k) Linear code with symbols from GF(2 m ). Let C 
denote the binary (nm,km) linear code obtained from C by representing each
code symbol by a m-tuple over GF(2) using the basis 16 1,a 2, ... ,Bm } f,-r
GF(2 m ). Let H be an (n-k)xn parity-check matrix of C. By rearranging the
-2-
A
bit positions, a parity-check matrix for the binary code C  can be repre-
sented in the following form:
b	 (1)H	 lH:B2H: ... :s mH) .
which is an (n-k)xmn matrix over GF(2 m ). For convenience, we will use the
order of bit positions given by (1). Let v-(vl,v2, ... ,vm ) be a binary vector
of mn components, where 
v
i (vil'vi2" .. 'vin) is a binary n-tuple for
1<i<m. Then, v is a codeword in C b if and only if
M
T
^ i Hv i
 = 0	 (2)
i=1
A.	 1
Let C denote the dual code of C. We assume that C does not contair.
the all-one vector (1,1,...,1). Let C e denote the linear code over GF(2m)
whose parity-check matrix is of the following form:
1 1	 1
H 
e 
=
	
	
(3)
H
Clearly C  is a subcode of C. Let C  and C e,b denote the binary sub-
field subcodes of C and C  respectively. Then C e b is the even-weight
subcode of Cb.
Let A,,(X) = AO_^+AO1 X+A02 X 2 +...+A O n X n be the weight enumerator of
C b . Then, AO i is the number of codewords of weight i in C b . Note that
A00-1' Ascuii ,e that there are k types of cosets modulo C  including C 
itself, and cosets of type-j have the same weight enumerator A A X) for
0<j<Z. Let Y be a (n-k)-tuple over GF(2m ). Then Y is said of 'type-j' if
and only if Y is the syndrome of a coset of type-j. Since C e b is the even-
weight subcode of C b . Each coset of C  can be partitioned into two cosets
of Ce,b , an even-weight coset and an odd-weight coset. Hence there are U
-3-
-4-
C--°)
types of cosets modulo 
Ce,b' 
Let A. (X) and A j,o (X) denote the evenA. ,e
part and odd part of A j (X) respectively, for 0<j<i.
^-1
For nonnegative integers s 1 , s 2" "' s k-1 such that ^ s < m, letj_l j —
Ns 
1 ,s 2
 „'s	 denote the number of (^1,y2,...,Ym)'s such thatR-1
(i) y, is an (n-k)-tuple over GF(2 m ) for 1<i<m;
(ii) the number of components Y i of type-j is s  for 1<j<T; and
(iii) the following equality holds
M
6 i'Y = 0
i=1
Then,	 .ollows from (2), (4) and the definition of N
	 that we
sl,s2,... ► sq,-1
have Theorem 1.
Theorem 1: The wei g ht enumerator of Cb , denoted Ab (X), is given by
b	 m-^ S.? (X) _	
Ns ,s ,...,s	
[AO(X))
	 Ajl(X)	 (5)
S 	 2	 R-1	 3-1
where S.
m
 = ((s1,s2,...,s^-1): s.^0(1<j.-O and L s . < m) and	 _	 S.
	
j=1	 j=1 J Al^
Let V=(v 1 ,v21"
 " v m ) be a binary vector of mn components where
v 
-(v il l v i2''" ' v i n) is a binary n-tuple for 1<i<m. Let C  be the bina*y
code of length mn derived from C by representing each code symbol of C
	
e	 e
by a binary m-tuple using the basis {B1'e2""' 5 m ). Then v is a codeword in
C  if and only if
e
m	 n
L Bi L vij = 0	 (7)
	i=1	 j=1
M
L ^.HGT = 0	 (8)
i=1
Since ^1,e2,...,Bm are linearly independent over GF(2), we have that
n
	
Lv.. = 0,
	 for 1<i<m .	 (9)
j=I
Hence we have Theorem 2.
(4)
fb
Theorem 2: The binary Code C  is an even-weight code and its weight enu-
merator Ab (X) is given by
Z-1	 S.
Ab (X) _	 N	 [A(X)
]m-^ E A. 3
 M ,
e	
S	 s1,s2, ... 1sk-1	
0,e	 j-
	
1	 (10)
	
Z-1
	
Z-1
where S Z'm	 {( s l , s 2 ,
 
.... s '-1 ): s an for 1<j<Z and
	 I s , < m) and a = I S.
	
J	 j=1 J	 j=1 J
A6
Let 
Cex 
denote the extended code obtained from C by adcing an overall
parity-check symbol. Hence C
ex 
is a code of length n+1 with symbols from
GF(2m ) and parity-check matrix
1 1	 1 1
0
	 ,M
H	 =	 H	 (11)
ex
0
Let 
Cex,b be the subfield subcode of CeX . Then `ex b is the extended
code of Cb . It follows from Theorem 1 that we have Theorem 3.
Theorem 3: The weight enumerator A eX (X) of Cex is given by
Q-1 S.
A  M _
	 Ns ,s ,...,s	 [AO,ex(X)]m-^ T( A]^ex(X)	 (12)
SZ'm	 Q-1	 j=1
R-1	 Q-1
where S^ 'm = {(sl,s2....,sR-1):s >0 for 1<j<Q and 	 s <m),	 _ I s . , and
	
J	 j=1 J	 j =1 J
A.	 (X) = A.	 (X) + XA.	 (X)	 (13)
J, eX	 J,e	 J'o
for 0<j<Q.	 AA
From Theorems 1, 2 anO 3, we see that, if we know the weight enumerators
of cosets of the binary subfield subcode C  and coefficients Ns ,s
	 ..,s	 '1 2	 Q-1
we can obtain the binary weight enumerators Ab (X), Ab (X) and AeX (X). weight
e
enumerators of cosets for some classes of codes are known, e.g., the Hamming
codes (2]. Let A  M denote the weight enumerator of a Hamming code which
is known (1-4]. Let C  be a Hamming code of length n=2 m-1. Then the
weight enumerator ACi of a coset of C  (other than C b ) is given by
ACH (X) = 
n
{(X+1) n - A H (X))	 (14)
-5-
	 in
r. v
J!
r"U"-K
	 r— 
4. 4 W W___ - -	
r9
If C  has minimum weight at least 2t+1 and all cosets of C  with
minimum weight t have the same weight enumerator A t (X), then it follows from
Macwilliams equation (2,51 that
n
At (X) = (n) -1 2- (n-k)	 L AJ pt (j) (1+X) n-j (1-X) jj =0
where A: is the number of codewords of weight j in the dual of C b and
J
P t (j) is a Krawtchouk polynomial. Theorem 4 provides a sufficient condition
for all cosets with the same minimum weight to have the same weight enumerator.
Theorem 4: If C  has minimum weight at least 2t+1 and the number of non-
zero weight w's such that there exists a codeword of weight w in the dual code
of C  is not greater than t+l, then the minimum weight of a coset other than
C  is at most t and all cosets of C  with the same minimum weight have :he
same weight enumerator.
Proof: In a coset of C b , there is at most one vector whose weight is not
greater than t. Hence this theorem follows immediately from Theorem 20 in
(p. 169;21.	 M
For example, the condition of Theorem 4 holds for primitive BCH codes of
minimum distance 5 and code length 2 m-1 with odd m A .
3. Binary ei g ht Enumerators for Some Reed-Solomon Codes
In this section we will derive the weiyht entunerators for the binary codes
obtained from some Reed-Solomon codes with symbols from GF(2 m ). Let C be a
Reed-Solomon code of length n=2 m-1 with generator polynomial g(X). Let u be
a primitive element of GF(2m).
Case 1 •	g(X) = X-C(.
In this case, the parity-check matrix for C is
-6-
(15)
t,
"I
The binary subfield subcode C  of C is the ramming code of length 2m-1.
There are two types of cosets of C  with weight enumerators A H (X) and
ACH (X) respectively. AH (X) is the weight enumerator of C b . ACH (X) is
the weight enumerator for the cosets with minimum weight equal to 1, and is
given by (14).
For v=(v 11v21 ... ,vm )EC b , v i belongs to a coset with weight enumerator
ACH if and only if 'Y = Hv110. Then. Ns with 0<s<m is equal to the
number of (Y1,Y2,•••,Ym)'s with s nonzero components for which
m
i=1 S iYi
=0
 .
Hence, N
s 
is the -ame as the number of codewords of weight s in a maximum
distance separable code of length m and minimum distance 2 with symbols from
GF(2m ). Consequently, we have (1,2)
s- 2
	
Ns = (5)	 (-1) 3 (s) (2m(s-j-1)-1)
j=0
Case 2:	 g(X) = (X-1)(X -a).
In this case, C
e 
has minimum distance 3. It follows from Theorem 2 that
m
A b M = I Ns[AH^eWIM-s[ACH^eWIs
S=O
where Ny is given by (16), AH a and ACH e are the even parts of A  an;)
ACH respectively. From Theorem 3, A 	 can be obtained.
Case 3:	 g(X) = (X- n)(X _OL
In this case, C has minimum distance 3 and
1 a a
2 	 an-1
H	 1	 a2 Cc	 C( 2(n - 1)
	 (18)
The binary subfield subcode C  is the Hamming code of lenqth 2 m-1. For
v=(v ,v,,,...,v ), let
1	 m
-7-
(16)
(17)
Yi1	
-T
Hv.	 1<i<m
Y i 2
Since v 
i 
is binary, we have
2
Y i 2	 Yi 1 .
Then v is a codeword in C b if and only if
Cm
L ^iYil	
0
i=1
M	 2
^.Y•	 = 0 ,i it
i=1
Note that v i is in a coset with weight enunerator ACH W if and only if
Y1l ^0. Since
cm
L ^Jil = 0
if and only if
M
C 2 2
L aiYil
i=1
N s is equal to the number of m-tuples, H1,S2,...,dm), over GF(2m ) with s
nonzero components for which
m
0
i=1	 (22)
M 2
	
0 .
i=1
Since, for 1<i<j<m,
1 ^0,
a2 B2i j
-8-
C4
v
(19)
(20)
(21)
i'
n
r	
A	 ,
Ns is equal to the number of codewords of weight s in a maximum distance
separable code of length m and minimum weight 3, and is given by (1,21,
Ns = (S) 513 (-1)3(.) (2m(s-j-2)-1) . 	 (23)
j=0
Then it follows from Theorem 1 that
m
Ab (X) = ^ N s [AH (X)]
m-s [ACH (X)] s	(24)
S=O
4here Ns is given by (23).
Case 4: g(X) = (X-1)(X-a)(X-a2)
In this case, C
e 
has minimum distance 4. It follows from Theorem 2 that
m
A (b) (X) _ ^ Ns[AH,e(X)]m-s(ACH,e(X)1s
S=O
where Ns is given by (23). Also, it follows from Theorem 3 Oat A  (X)
can be obtained.
For all the cases considered above, the binary weight distribution is
independent of the choice of the basis {6 1 ,B2'" ''Sm)'
Case 5: g(X) = (X-a)( X -a3), or (X-a)(X-a2)(X-a2)(X -a3) or
(X-00(X-a2)(X-a3)(X-a4)
In either case, C  is the primitive BCH code of length 2 m-1 and min-
imum distance 5. Hence C  is quasi-perfect (2-41. For odd m. C  satis-
fies the conditions of Theorem 5, and there are three types of cosets of C 
other than C  with minimum weights 1, 2, and 3 respectively. The weight
enumerator A R (X) for 1<Z,<2 can be obtained by MacWilliam's equation given by
(15), and A 3 (X) is given by the following equation:
A
3 (X)= [2n - 2k(1+n+(2))]-1{(X+1)n - A 0 (X)
nA l (X) - (n
2 )A2
(26)
(25)
-9-
PER
Consider the case for which g(X)-(X -a)(X -0( 3).	 For v-(vl,v2,...,vm)
with v i
 as a binary n-tuple for 1<i em, lot
FY:1
	
Ly	 Hv i.13
Then, v is a codeword in C b if and only if
c
m	 m
iL1SiYil - 0
	 and	
1Z1aiYi3 = 0 .
For 1<i<m, v i is a codeword in C  if and only if Yil:.Yi3-0; v i is in
a coset with minimum weight 1 if and only if 'i3 Ii1A0; v i is in a coset
with minimum weight 2 if and only if (il00 and trace (l+Y 13/)il)70; and
otherwise v i 
is in a coset with mini mum weight 3. A closed formula for
N	 is under study.
sl,s20s3
	
cases are:	
-1	 2	 -1	 -2
Other interesting	 g(x)=(x-a)(x-a ) or(x-a)(x-a )(X-a )(X-a ).
There exists a cyclic code with the
those of the extended code C eX . For
the binary subfield subcode C 	 of
ex,h
Zetterberg's code [2,5) f r even m.
same n, k and the minimum distance as
the case with g(X) _ (X-a)(X-a 1),
the cyclic version of C eX is a
However, the weight distribution of a
coset of Cex b is unknown.
4. Conclusion
In this paper, we have investigated the weight distribution of binary
linear block codes derived from codes with symbols from GF(2m ). We'ght
enumerators for binar y
 codes derived from some Reed-Solomon codes over GF(2m)
have been obtained.
Reed-Solomon codes with syinrols from GF(2 m ) are widely used as the
outer coder in a concatenated coding scheme for error control in data communi-
cation. Recently, we are investigating a concatenated coding scheme for
NASA's Telecommand System. Two possible outer ;odes are considered, one is
the X.25 standard code with generator polynomial g(X) = X 16+X 12 +X 5 +1 and
-10-
IN
the other is the Reed-Solomon code with symbols from GF(2 8 ) and generator
polynomial g W ={X-1)(X-o). The case with X.25 standard code as the outercode
has been analyzed. Now we are analyzing the case with the a')ove Reed-Solomon
code as the outer code. Knowing the binary weight distribution of the
Reed-Solomon code, we should be able to analyze the performance of the
propose9 concatenated coding .scheme for NASA's Telecommand System.
-11-
J]
-12-
6
REFERENCES
1. S. Lin and D.J. Costello, Jr., Error Control Coding: Fundamentals and
Applications, Prentice-Hall, New Jerse y , 1983.
2. F..:. Macwilliams aod N.J.A. Sloane, Theory of Erro:• • Correcting Codes,
North Holland, Amsterdam, 1977.
3. E.R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968.
4. W.W. Peterson and E.J. We'-don, Jr., Error-Correcting Codes, 2nd. ed., MIT
Prejs, Cambri6ge, Mass., 1972.
5. F.J. MacWilliams, 'A Theorem on the Distribution of Weights in a
Systematic Code, Bell System Technical Journal, Vol. 42, pp. 79-94, 1963
6. L.H. Zetterberg; ` Cyclic Codes from Irreducible Dolynomials for Correc-
tion of Multiple Errors,' IEEE Transactions on Information ThPoFy, Vol.
8, pp. 13-20, 1962.


On the binary weight distribution of some Reed-Solomon codes

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