DESCRIPTION OF BINARY I~EED-MULLER CODE [Kasami, Lin, and. Peterson (1968) and Berlekamp (1968)]. DEFI~ITm~
INFORMATION ANJ) CONTaOL 14, 442-456 (1969)
Restrictions on Weight Distribution of Reed-Muller Codes E. R. BERLEKAlV/P ANn N. J. A_. SLOANE Bell Telephone Laboratories, Murray Hill, New Jersey I t is s h o w n t h a t in the r t h order b i n a r y R e e d - M u l l e r code of l e n g t h N = 2 " a n d m i n i m u m distance d = 2~-L t h e only codewords h a v i n g weight between d a n d 2d are those w i t h weights of t h e form 2d -- 2 t for some i. T h e same r e s u l t also holds for c e r t a i n supercodes of t h e R M codes. 1. D E S C R I P T I O N
OF B I N A R Y I ~ E E D - M U L L E R C O D E [Kasami, Lin, and P e t e r s o n (1968) a n d B e r l e k a m p (1968)]
DEFI~ITm~ 0, be any power series of the form P(~) =
E
kEK(n)
a kz ' = k
i~l
a~z k~.
(Sa)
Clearly any power occurring in P(~) is of the form 2 ~ -- 1 + ~
&(n),
for s o m e j .
(9)
T h u s S ( z ) is of the form of p(~,--r). Power series of this t y p e will play an i m p o r t a n t p a r t in the proof of the theorem. We say t h a t P(~) has degeneracy of order ~r, ~: >= O, if ak~ = 0 for 1 i _-< ~r, ak~+~ ~ 0. I n this case we define A Zr+l
BERLE~P
446
AND SLOANE
TABLE I BINARY EXPANSION
OF ELEMENTS
k l , ]~2 , "*" OF
K(n) IN
INCB.E&SING
ORDEI~
Elements
Length n
1~01~-~,
n+l
1 -< r--< n
ln+l
1"01~01~-~-~1 -< s -< n 1"01~-s+1 JO -- r ~ n I~+I0
DIFFERENCES
~,(n)
n+2
} s
--
TABLE II , i = 1, 2, .--
=
]gi+l - - k i
, BETWEEN
ELEMENTS
OF K(n) 2"-1 2~-1
2 ~-: 2~-z 2~-~
2~-~ 2~-a 2 ~-a 2 ~-3
... .....
2 2 2
1 1 1
1 1 1
.."
2
1
1
2
1 1
1 1
,.,
1 1
I t f o l l o w s t h a t g(~' ~) is a n y p o w e r series w i t h n o n z e r o c o n s t a n t t e r m a n d d i f f e r e n c e s b e t w e e n s u c c e s s i v e p o w e r s g i v e n b y T a b l e I I w i t h ~r i n i t i a l t e r m s r e m o v e d , i.e. b y T a b l e I I I . F o r m a l l y , l e t ;0
d~(n,
if
0 = j > 1. Their formulas are then as follows: I f j = 2 or max(a, 2) < j =< b, then r+2j~3
N~,~,~ = 2 ~-~°+'(s+"
r--3 H i~O
II (2~-~ -- i) ~=0 _ _
(1)
3"--1
(2r-2-i- - i)"H (z~i+I- - I) i~O
If max(b, 2) < j _-< a, then 3r--j--1
N .....
2 ++3)-~-~
=
II i=0 ~-j-1 (2 r - '"- '" -
( 2m-~ -- 1) (2) 1) •
(2 r-¢ -- 1)
Li=or~
]~"
If 3 =< j _-< min(a, b), then N ...... is equal to the sum of (1) and (2). I n August 1968 we obtained the same results for the special case of 2nd order Reed-Muller codes by another method. R~CEIVED: J a n u a r y 6, 1969; revised March 28, 1969. REFERENCES BEI~LEKAMP,F. R. (1968), "Algebraic Coding Theory," McGraw-Hill, New York. KASAMI, T., L~r~, S., AnD PETEI~SO~,W. W. (1968), New Generalizations of the Reed-Muller Codes--Part I: Primitive Codes, IEEE Trans. Inform. Theory, IT-14, 189-198. KASA•t, T., To~:v~A, N., AND HaTa~-ax~, S. (1969), On the Weight Structure of Reed-Muller Codes, papers Tech. Group Inf. Theory, I.E.C.E., Japan McElieee, R. J. (1967), Linear Recurring Sequences Over Finite Fields, Ph.D. Thesis, California Institute of Technology.
