Restrictions on Weight Distribution of Reed-Muller Codes - Neil Sloane

21 downloads 920 Views 484KB Size Report
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.