INFORMATION AND CONTROL 5, 153--162 (1962)
On the Number of Information Symbols in
Bose-Chaudhuri Codes*
HENRY B. MANN
Department of Mathematics, Ohio State University
Let a be a primitive root of G. F. (q~). Let I(m, v) be the number
of information symbols of the code with parity check matrix (a*~),
i = 1, -.. , v , j = 0 , - " , q m - 2. Let v = qX, m _ ), =r. Thenfor
sufficiently large m we have
l(m,
v) =
(p'°>
where (c> denotes the nearest integer to c and p is the positive root
of the e q u a t i o n x ~ = ( q - - 1) (x~-1 + ... + 1). For small values of
m we have I(m, v) = p ' ~ - ewhere t~] -< ( r - - 1)r m, [r ] < 1. Estimates for r are given in the Appendix. Two other formulas, exact
for all values of m _>- r - 1, are also given. The first contains r terms,
the second [m/r ~ 1] terms, where [c] denotes the largest integer
not exceeding c.
L I S T OF SYMBOLS
G . F . = G a l o i s field
(x} = n e a r e s t i n t e g e r t o x
Ix] = l a r g e s t i n t e g e r n o t e x c e e d i n g x
T h e c o d e s in t h e t i t l e w e r e i n t r o d u c e d b y B o s e a n d C h a u d h u r i ( 1 9 6 0 ) .
A n e x c e l l e n t a c c o u n t of t h e m , i n c l u d i n g t h e n e c e s s a r y b a c k g r o u n d i n
a l g e b r a h a s b e e n g i v e n b y P e t e r s o n (1960, 1961) a n d t h e r e is n o n e e d
to g i v e a d e t a i l e d d e s c r i p t i o n of t h e m h e r e .
L e t a b e a p r i m i t i v e r o o t of G . F . (qm). L e t H ( x ) b e t h e p o l y n o m i a l
of s m a l l e s t d e g r e e w i t h c o e f f i c i e n t s in G. F . (q) w h i c h h a s a, a ,2 . . . , OLv
a m o n g i t s r o o t s . T h e d e g r e e h of H ( x ) is t h e n u m b e r of c h e c k s y m b o l s
of t h e c o d e w i t h p a r i t y c h e c k m a t r i x ( d 3 ' ) , i = 1, • - • , v, j = 0, - • • ,
qm--2.
T h e n u m b e r I ( m , v) of i n f o r m a t i o n s y m b o l s is g i v e n b y
Sponsored by the Mathematics Research Center, U. S. Army, Madison, Wisconsin, under Contract No. DA-11-002-ORD-2059.
153
154
MA~N
I ( m , v) = q,n _ 1 -- h. For q = 2 P e t e r s o n (1960) p r o v e d I ( m , v ) =
o(2 m) for v / 2 '~ > c > 0. W e shall i n this p a p e r give two exact f o r m u l a e
for I ( m , v) w h e n v = qX. One of our f o r m u l a s i m m e d i a t e l y implies
P e t e r s o n ' s result.
•
qra--l£
I f a ~is a root of H ( x ) t h e n a qj, a q2j, • . . , a
are also roots of H ( x ) .
O n the o t h e r h a n d t h e e x p o n e n t s of a can be considered mod. qm _ 1.
H e n c e g i v e n v we h a v e to find t h e n u m b e r of residues y mod. qm _ 1
which satisfy a c o n g r u e n c e
q"y = x ( m o d , qm _ 1)
for some 0 < g < m - 1 a n d some 1 --- x --- v. If we write the residues
y i n t h e n o t a t i o n to t h e base q t h e n qy can be o b t a i n e d from y b y a
cyclic p e r m u t a t i o n of its digits. W e therefore h a v e the following l e m m a :
LEMMA 1. The degree of H ( x ) equals the number of integers y such that
0 < y < q'~ -- 1 and such that an intger y' <= v m a y be obtained f r o m y by
permuting cyclically the digits of y, when y is written i n the notation to the
base q.
W e n o w a s s u m e v = qX. A n u m b e r
y = Co -t- clq 4- "'" -4- cm-lq
m--i
is less t h a n v = qX if a n d o n l y if c~ . . . . .
cm_l = O. H e n c e we h a v e
LEMMA. 2. I f V = q× and m -- h = r, then the degree of H ( x ) is one less
than the number of sequences of length m of q digits O, 1, . . . , (q -- 1),
which i f placed on a circle contain a r u n of r or more O's.
( F o r e x a m p l e t h e sequence 010100 c o n t a i n s a r u n of 3 zeros if placed
o n a circle.) F r o m here o n we shall drop t h e r e s t r i c t i o n t h a t q is a p r i m e
power, h A- 1 will d e n o t e the n u m b e r of sequences of l e n g t h m of q
digits 0, 1, • • • , q -- 1 which c o n t a i n a " c i r c u l a r " r u n of a t least r zeros.
O n e can reduce the e n u m e r a t i o n of these " c i r c u l a r " r u n s to a n e n u m e r a t i o n of r u n s i n the o r d i n a r y sense, which we shall call s t r a i g h t r u n s .
Let r be fixed a n d let l(s) be t h e n u m b e r of sequences of l e n g t h s which
c o n t a i n a s t r a i g h t r u n of l e n g t h r. P u t h -t- 1 -- l ( m ) = d ( m ) . T h e n
d ( m ) is the n u m b e r of sequences of l e n g t h m which c o n t a i n a circular
r u n b u t no s t r a i g h t r u n of l e n g t h r. I f 2r = m t h e n such a sequence
m u s t c o n t a i n a circular r u n of l e n g t h r + k, 0 <= k < r - 2, b o r d e r e d
b y a n o n z e r o digit o n b o t h ends, while the r e m a i n d e r of the sequence
does n o t c o n t a i n a s t r a i g h t r u n of l e n g t h r. M o r e o v e r the r -[- k zeros
m u s t be placed i n such a w a y t h a t no s t r a i g h t r u n of l e n g t h r results
INFORMATION SYMBOLS IN BOSE-CHAUDHURI CODES
a n d this can be done in r -- k -
155
1 ways. H e n c e for m > 2r
h -t- 1 = l ( m )
+ ( q - - 1) 2 ~ ( r - - /c-- 1)(q . . . . k - 2 _ l ( m - - r - - /c-- 2)).
k~0
W e now p u t ~ ( s ) = q* - - l ( s ) T h e required n u m b e r z(m) of sequences
of l e n g t h m w i t h o u t a circular r u n of l e n g t h r is t h e n g i v e n b y
r--2
z(m) = ~ ( m ) - ( q - 1 )
~z(r-k-1)~(m-r-l~-2).
(2)
k~0
W e proceed to c o m p u t e l ( s ) . A sequence $1 of l e n g t h s, s > r c a n be
o b t a i n e d from a sequence S of l e n g t h s - 1 b y a d d i t i o n of a n y of the
q digits 0, 1, . . . , q - 1. If $1 has a r u n of l e n g t h r while S does n o t
h a v e such a r u n t h e n $1 m u s t h a v e its last digit 0 a n d S m u s t e n d w i t h
(r - 1) zeros preceded b y a n o n z e r o digit. Moreover, S c a n n o t have
a r u n of r zeros i n its first s - r - 1 digits. H e n c e
l(s)
=: q l ( s -
l(r)
=
1,
1) + ( q -
1)(q'-r-~-
l(s-
0
l(]~) =
1)),
r-
forO
s > r,
(3)
~ k < r.
Hence
1) -
~(s) = q ~ ( s ~o(r) = q r _
1,
(q-
i)~(s-r-
~(k) = q
k
1),
s>r,
f o r 0 < k < r.
(4)
I t is well k n o w n t h a t we m a y p u t ~(s) = p~. T h e n
pr-t-1 - q K + q -
1 =o.
(5)
F o r r = i we o b v i o u s l y h a v e ~(s) = q - 1. H e n c e we m a y assume
r > 1. A sketch of the curve y = x ~+~ - - qx ~ -t- q -- 1 = 0 shows t h a t
for r > 1 (5) has no double roots. Let n o w 1, p~, • • • , p~ be the roots of
(5). W e m a y p u t ~,(s) = co + c~m ~ + • • • -t- c,p, ~. T h e i n i t i a l c o n d i t i o n s
of (4) give
Co+
c~p~ k--i- " "
+
Co+
c~p/ -t- " ' " +
c~m k = qk,
0 =< lc < r
(6)
c ~ p f = q~ - - 1.
W e divide (5) b y p -- 1, p ~ 1 a n d get
p~ = (q -- 1)(p'-~ +
. . - -[- 1).
(7)
156
~A~N
S u b s t r a c t i n g the sum of the first r equations of (6) f r o m the last
a n d using (7) we find co = 0 a n d the last e q u a t i o n of (6) m a y be deleted.
Deleting the last e q u a t i o n of (6) we see t h a t each c~, i = 1, • • • , r is
given b y ~ quotient of two V a n d e r m o n d d e t e r m i n a n t s and p u t t i n g
f(x) = x ~- (q1 ) ( x ~-1 ~- . . . + 1) one finds
II
(q -
~i
p~)
f(q)
-
c, = ~ I (P, -- P])
(s)
(q -- P~)f'(P~)"
Using (5) this can be simplified to
r
c~- -
P~
f f ( m ) ( q -- 1)
(9)
W i t h (x -- 1)f(x) = x ~+1 -- qx ~ -t- q -- 1 we easily find for p ~ 1 using
(5)
p(p-
1)f'(p)
= p~+l
1)r
(q_
and
p:.+~(p~ - 1)
c~ = (p~+~ _ (q _ 1 ) r ) ( q -
p,(p( 1)
r+l
p,
-
1)
(q -
1)r
}
i=
(10)
1, . . . , r .
Hence
q~(8)
A sketch
has besides
the interval
root of g ( x )
,
~=I
p,(p(
1)
pl -- ( q - - 1)r
-
(11)
of the curve g ( x ) = x ~+1 - qx ~ + (q -- 1) shows t h a t g ( x )
the root p = 1 only one other positive root w h i c h lies in
( ( r / r + 1)q, q). I f p is a nonpositive (negative or complex)
then f r o m (5) we get with r = I P I
r
r+l
> qr'--
(q--
1)
< qv ~ -- ( q - -
1).
and f r o m (7) for r > 1
r
r+l
Hence
LEMMA 3. I f p is a nonpositive (negative or c o m p l e x ) root of (5) then
Ip1<1.
INFORMATION
SYMBOLS
IN
BOSE-CHAUDHURI
157
CODES
W e shall now prove
L ~
4.
r-1
1
if
j c~p~ 1 < 2 ( r _ l )
p,J<l.
(12)
F r o m (11) we have
ic~l <
Ip, l l p , '
1]
-
ip~ [p,~- 1]
= (q -- ~)r 2_-1- =
r -- i
(13)
W e p u t p~ = re *~. W e m a y choose 0 < e =< ~-. F r o m (5) we get
r-}-I
T
cos (r -1- 1)e -~ q -- 1 = qT r COS (re),
(14)
r sin (r -t- 1)e = q sin (r~).
Substituting f r o m the second equation of (14) into the first we get
(q--
1) s i n ( r e )
= r'+lsine,
(15)
(q -- 1) sin (r -t- 1)e = rrq s i n e .
F r o m (14) and (15) we get
cos (re) > 0,
sin (re) > 0,
sin e > (q -- 1) sin (r~)
(16)
cosy < cos (re).
Now
I p,r
--
112
~
1
--
.r~T re o s ( r e ) +
T2r
.
I f s i n e = 0 t h e n cos (re) = 1 > r r. If s i n e ¢ 0 we h a v e f r o m (15)
(sin(re)
(q-1)\~n~
cos e + cos (re)
)
=q~.
I f cos e < 0 t h e n cos (re) > r ~ is i m m e d i a t e l y obvious. If cos e > 0
we establish the inequality f r o m the inequaliLies (16). H e n c e we h a v e
in all cases
c o s ( r ~ ) > r ~.
W e now get
(I p; II p: - 1 I) ~ < ~ ( 1
- ~)
<___
L e m m a 4 now follows f r o m (13). F r o m (12) we find
~_,
]p~[<l
cipi t < ½ for
t ~ r--
1.
158
M~NN
Denoting now b y {x} the nearest integer to x we therefore have:
THEOREM 1. The n u m b e r ~ ( m ) of sequences of length m >= r -- : of q
digits O, 1, . • • , q -- 1 which contain no straight r u n of r zeros is given by
~(m) = <cpm)
(17)
where o is the largest positive root of Eq. (5) and
c = p ( p ~ - - 1)/[o ~ + ' -
(q-
1)r];
moreover ( r / r + 1)q < p < q.
The last inequality follows from the fact that the polynomial
x ~+'-
qx r + q - -
1
has a minimum at x = ( r / r + 1)q where it is negative and the value
1 for x = q.
F r o m (2) we get the following corollary:
COnOLLA~Y 1. L e t z ( m ) be the n u m b e r of sequences of length m of q
digits O, 1, . . . , q m >_2r
1 which contain no circular r u n of r zeros. T h e n f o r
r--2
z(m)
(q -- 1 ) ~ ( r - -
= (cp '~) -
k - - 1)(cp . . . . . ~-2).
(18)
k=0
F r o m (7) and (11) we have
r--2
z(m)
~ci(p~
m
--
(q -- 1)~Y'~ (r -- k --
i
1 \
)mm - - r - - k - - 2 \)
0
r--2
and s i n c e f ' ( p )
= rp ~-1 -- (q -- 1) ~
(r -- k - 1)o r-k-2 this reduces to
b~0
z(m)
~-~ci(pl m + ( q -
X~et/
; ) [ p , ) p\ l m - - 2 r
(q-
1~
) r mm - - r - - l x).
(19)
i
Substituting for cl its value from (9) and (11) one finally finds
(20)
zOn) = E p,",
i
whence
CO~OLLA~Y 2. F o r m
z(m)
>= 2 r a n d
loci
<= .r < l f o r [ m [ < l
= p r o + e,
I el
< (r--
1 ) r m.
(21)
For sufficiently large m
z(m)
= (p~).
(22)
INFORM-~TION SYMBOLS IN BOSE-CHAUDHURI CODES
159
TABLE
VALUES OF p AND C p r FOR q = 2
r
p
cp r
2
3
4
5
6
7
8
9
10
11
12
1.618034
1.839287
1.927562
1.965948
1.983583
1.991964
1.996031
1.998029
1.999019
1.999510
1.999756
3.065248
7.077522
15.07028
31.05669
63.04252
127.0302
255.0206
511.0136
1023.009
2047.005
4095.003
COROLLARY 3. The number I ( m , q×) of information symbols of a code
with parity check malrix (a~J), i . . . . 1,
m-2, a a
, q~, j = O, 1 , . . . , q
primitive root of G. F. (qm) is given by
I = z(m),
(23)
where z(m) is given by (18) with r = (m - X). Within the given error
z( m) may o] course also be computed from (21).
Table I shows the values of p and cpr for q = 2 and 2 < r < 12.
In the applications to coding r is usually not large, so that (18) will
be convenient to use. If m is large compared to r, then Eq. (22) can be
used. We shall therefore give another formula for h + i which is a sum
of [m/r + 1] terms, so that it can be used when r is too large to use
(18) and m too small to use (22). We shall need the following lemma.
LEMMA 5. Suppose we have N sets oJ objects, say balls to fix ideas.
The sets containing either 1, 2, . . . balls. Let al be the number of balls,
as the number of pairs of balls that occur together in one set, 33 the number
of triples etc. Then
N--
al-
as+ a3....
(24)
PROOF: Number the balls. N~ = 1 is the number of sets containing
the ith ball, hr~j (either 0 or 1) the number of sets containing the ith
and j t h ball etc. T h e n N is the number of sets containing at least one
ball and b y a well known formula
N
=
=
...
al
--
a2
"-~
• ...
160
MANN
I f a sequence does n o t consist entirely of zeros and if it contains a
circular run of r zeros t h e n it m u s t contain a nonzero digit followed
(in a circular sense) b y r zeroes. W e shall call a nonzero digit followed
b y r zeros a string. W e can place such a string in m positions and then
fill the remaining positions b y digits in q . . . . i ways. H e n c e a l , the n u m ber of strings occurring in all sequences is mq ..... ~(q -- 1). T h e n u m b e r
ak of k-tuples (k(r -~- 1) -< m) of strings is o b t a i n e d b y placing the first
string in m positions. T h e other strings can now be considered as units
and can be distributed
i n ( m - k/~r -- 1 1) w a y s over the m - kr - 1
places still to be filled. T h e remaining places can be filled b y digits in
q~-ke+l) w a y s and the initial terms of the strings can be chosen in
(q -- 1) ~ ways. E a c h k-tuple has now been counted/~ times since each
of its strings can be considered the first. H e n c e
m
a~ = ~ ( q - -
1) k
(rn--kr--1)
k--
q,~-k(~+~).
1
B y (24) we therefore h a v e :
h=m
~
k=l
( - 1 ) ~-~ ( q - 1)~ ~ - k r - 1
k
k -
1
q~-~(,+,
(25)
and
z(m) = q " -
h -- 1.
(26)
I t m a y be observed t h a t all n u m b e r s
v, qX <= v < qX -I- q~-~ -t- "'" + qX-~-~/'J~ = n(X)
contain in their representation to the base q a sequence of at least r
zeros. Hence the n u m b e r I(m, v) of information symbols for the corresponding B o s e - C h a u d h u r i code will be z(m) as given b y (18). F o r m u l a
(21), (22), or (26) m a y also be used if more convenient. I t is also
obvious t h a t all n u m b e r s obtainable f r o m n(X) b y cyclic p e r m u t a t i o n
of its digits are distinct a n d larger t h a n n(X), so t h a t I(m, n(X)) =
z~ - m. Similarly one sees t h a t I(m, qX _ 1) = z~ - b m .
APPENDIX
Let re '~ be a nonpositive root of (5). W e m a y assume 0 < ~ < ~r.
P u t r f ~ a ( m o d 2~r.) F r o m (14) we h a v e 0 ==_ a < ~r/2, sin a < 1/q <
sin 0 r / 2 q ) , hence qa < ~r/2. F u r t h e r m o r e sin ( a + ¢) > q sin a >
sin qa and so
a + ~ > qa,
~ > (q
-
-
1)a.
(27)
INFORM~kTION
SYMBOLS IN
BOSE-CttAUDHURI
CODES
161
W e shall need the following L e m m a :
LEMMA 6. For 0 < kq~ <~ 7c/2, k > 1, the function
~(~) = sin ( k ¢ ) / s i n
decreases with ~.
One p r o v e s L e m m a 6 easily using the convexity of tan ~.
F r o m (27) we h a v e r~ ~ 21r, hence
> --.2~
(28)
r
If sin ~ = 0 we m u s t h a v e ~ = 7r and (14) gives
< (q -- 1 ) / ( q - k - 1).
F o r sin ~ ~ 0 we h a v e f r o m (15)
s i n a + ~ q -- 1
sin ~
q
"
T~
-
(29)
F r o m (27), (28), (29), and L e m m a 6 one easily establishes the
following inequalities:
r~< q-
1
if
v>~>
7r
2'
q
~<:q--1
q
T r
1
sin(q -- 1/q)
if
~r >
> q-- 1~
2---~=
q
2'
q --- 1 sin [(q/q -- 1)2at/r]
q
sin(2~r/r)
~
if
-
q--lit
~ < - - q
2"
This gives the following e s t i m a t e s for T~:
('q--
1~ ~/~+1
r" < \ q ~ - - ~ ]
for
r~<q--
q
1
for
~ ---- It,
2--< r < 4,
r" < q -- 1
1
q
sin(2~r/r)
for
4 q ~ r -> 4,
q -- 1 --
r~ < q -- 1 sin [(q/q -- 1)2~r/r]
q
sin(27r/r)
for
r >=
4q
q--l"
(30)
162
~ANN
F o r q = 2 ~ short calculation shows t h a t for r = 2, 3, 4, 5, 6 Eq. (22)
is exact for m _-> 2, 6, 10, 18, 24, respectively.
RECEIVED: October 9, 1961
REFERENCES
BOSE, l~. C., AND I~AY-CHAUDHURI,D. K. (1960). On a class of error correcting
binary group codes. Inform and Control 3, 68-79.
BOSE, R. C., AND RAr-C~AuDHURI, D. K. (1960). Further results on error correcting binary group codes. Inform. and Control 3, 279-290.
PET~SON, W. W. (1960). Encoding and error correction procedures for the BoseChaudhuri codes. IRE Trans. on Inform. Theory IT6, 459-470.
PETERSON, W. W. (1961). "Error Correcting Codes." M I T Press and Wiley,
New York.