Corollary 9.3.3 gives an easy recursive method for calculating the Krawtchouk coefcients, with Corollary 9.3.2(1) and (3) providing
initialization.
The proposition allows us to reformulate MacWilliams Theorem as
(9.3.4) Theorem. (MacWilliams Theorem in Kr
!
Km(i;n,s)ai 0. i=0
In view of Theorem 9.3.4, Delsartes Theorem can be thought of as a nonlinear version of
MacWilliams Theorem. Our proof here of Delsartes Theorem follows Simonis and DeVroedt
(1991). For linear codes A we also recover MacWilliams Theor
and (W |v) = 0, as required. P (9.4.5) Corollary. Suppose that, for some set of constants u,
!
u (u|v) = 0 , uV
forall 0=vV. Thenu =isconstant,foralluV.
Proof. By Lemma 9.4.4(1), a constant choice u = does have the stated property. In particular, after su
9.4. LLOYDS THEOREM AND PERFECT CODES 139
An F-linear code of length n will be, as before, any nonempty subset C of Fn that is closed under addition and scalar multiplication. The
code C dual to C is also dened as before:
C =cfw_vFn |vc=0, forallcC, and i
9.4. LLOYDS THEOREM AND PERFECT CODES 141 (9.4.4)Lemma. LetvV.
(1) Always
(V|v) = |V|ifv=0 = 0 otherwise.
(2) If W is an F-linear code in V, then
(W|v) = |W| if vW
= 0 otherwise.
Proof. If 0= v V , then by Property (ND) of Lemma 9.4.3 there is a wordaV wi
9.3. DELSARTES THEOREM AND BOUNDS 135 Proof. The rst Delsarte inequality yields
n
J
for each m-subset J.
(9.3.9) Lemma.
[m]
c
=
!
cj , J
jJ
!
(n 2i)ai
0
!
= n+ (n2i)ai
i=d
n
!
n+(n2d) ai
i=d
= n+(n2d)(|A|1)
= n(n2d)+(n2d)|A|
= 2d+(n2d)|A|.
This implies (
9.2. MACWILLIAMS THEOREM AND PERFORMANCE
Lemma 9.2.3 and the previous calculation now give
129
PR = =
=
=
=
=
1PD
M
j=1
1
1!
Prob(sj(e)= 0|j)
2nk1 M
1
1!
(1 (q p)wj )/2
2nk1 M
j=1
1
1!
(1(qp)wj)
2nk
M
M
1! 1!
1 nk 1+ nk
2
2
j=1 j=1
1!
(q p)wj
2nk
M
j=1
(9.2.1) Proposition. Let PD be the probability of detecting an error, PE the probability of false decoding, and PR the probability of getting a
decoding result.
(1) PR = qn + PE.
(2) PR + PD = 1.
!
(3) PR = ni=0 ci qnipi = WC (q, p).
Proof. The rst two pa
Chapter 9
Weight and Distance Enumeration
The weight and distance enumerators record the weight and distance informa- tion for the code. In turn
they can be analyzed to reveal properties of the code. The most important result is MacWilliams
Theorem, which
As our proof of the Plotkin bound in Corollary 9.3.8 suggests, these methods can be used to nd general bounds; but new bounds of this
type are very difcult to prove. On the other hand, the linear programming bound is remarkably effective in specic cases,