Báo cáo toán học: "MacWilliams identities and matroid polynomials" doc

MacWilliams identities and matroid polynomialsThomas Britz Department of Mathematical Sciences, University of Aarhus, Denmark britz@imf.au.dk Submitted: December 12, 2001; Accepted: Apri

MacWilliams identities and matroid polynomials

Thomas Britz

Department of Mathematical Sciences, University of Aarhus, Denmark britz@imf.au.dk Submitted: December 12, 2001; Accepted: April 22, 2002

MR Subject Classifications: 05B35, 94B05

Abstract

We present generalisations of several MacWilliams type identities, including those by Kløve and Shiromoto, and of the theorems of Greene and Barg that describe how the Tutte polynomial of the vector matroid of a linear code determines therth

support weight enumerators of the code One of our main tools is a generalisation

of a decomposition theorem due to Brylawski

Since the 1963 article [8] by F J MacWilliams, coding theorists have paid considerable attention to the support (Hamming) weight distribution of linear codes In later years,

this interest has increased due to results such as those by Wei [16] on rth generalised Hamming weights, Kløve [6] and Simonis [13] on rth support (Hamming) weight

distri-butions (effective length distridistri-butions in Simonis’ terminology), and Shiromoto [11] on

λ-ply weight enumerators Section 2 of this paper introduces notation and the various

enumerators, by presenting the MacWilliams identities [8] as well as their generalisations

by Kløve [6] and Shiromoto [11] The two main results of this section, Theorems 3 and 7, generalise these results Proofs of these theorems appear in the later sections

In Section 3, we generalise theorems due to Greene [5] and Barg [1] that describe

how the Tutte polynomial of the vector matroid of a linear code determines the rth

support weight enumerators of the code We obtain two theorems which turn out to be equivalent to each other and to the ‘Critical Theorem’ by Crapo and Rota [4] The main tool is a generalisation of the characterisation of Tutte-Groethendieck polynomials due to Brylawski [3] As applications of these theorems, we prove Theorems 3 and 7 of Section 2

In Section 4, an alternative proof of Theorem 7 is presented This proof relies on coding-theoretical arguments rather than on matroid theory

We assume a basic knowledge of matroid theory; for an excellent introduction to the topic, see [10, 17, 18]

2 Support enumerators of a linear code

{f e | e ∈ A} whose elements f e are labeled by the elements e of A ⊆ E A linear code

E

q If v = {v e } E is a word of F

E

q, then

E

of v.

weight enumerator

A(z) =

n

X

i=0

A i z i

fundamental identity between the support weight enumerators of a linear code and its dual

Theorem 1 (MacWilliams identity) [8] If A(z) and B(z) are the support weight

enu-merators of a linear k-dimensional code C ⊆F

E

q and of its dual C ⊥ , then

B(z) = 1



.

given by

A {z e } E

E 0 ⊆E

A E 0

Y

e∈E 0

z e

MacWilliams identity for support enumerators A proof will be provided in Section 4, but for now remark that it follows from an equivalent result, Proposition 2 in [14], or from

Theorem 2 Let C ⊆F

E

q be a k-dimensional linear code If A {z e } E

and B {z e } E

are the respective support enumerators of C and the dual code C ⊥ , then

B {z e } E

q k

Y

e∈E

1 + (q − 1)z e


An 1− z e

o

E



.

A further generalisation of the support weight enumerator is the m-tuple support

enu-merator

A [m]({z e } E) = X

E 0 ⊆E

A [m] E 0

Y

e∈E 0

z e

where A [m] E 0 denotes the number of ordered m-tuples of codewords in C whose union of

is as follows

Theorem 3 If A [m] {z e } E

and B [m] {z e } E

are the m-tuple support enumerators of a linear k-dimensional code C ⊆F

E

q and of its dual C ⊥ for some m ≥ 0, then

B [m] {z e } E

q km

Y

e∈E

1 + (q m − 1)z e


A [m]n 1− z e

1 + (q m − 1)z e

o

E



.

We will prove this result in Section 3

Corollary 4 For each subset E 0 ⊆ E it holds that

X

E 00 ⊆E 0

B E [m] 00 = (q m)|E 0 |−k X

E 00 ⊆E\E 0

A [m] E 00

Proof Set z e = 1 for each element e ∈ E 0 and z e = 0 for each element e / ∈ E 0 Now

E 0 :|E 0 |=i

A [m] E 0 for i = 0, , n.

Let the m-tuple support weight enumerator of C be

given by the sum

A [m] (z) =

n

X

i=0

A [m] i z i

As an im-mediate corollary of Theorem 3, we obtain the following generalisation of the MacWilliams identity by K Shiromoto

Theorem 5 [11]

k-dimensional code C ⊆F

E

q and of its dual C ⊥ for some m ≥ 0, then



.

A different generalisation of the support weight enumerator of a linear code C involves

v∈C 0

S(v) = i} The rth support weight enumerator is the corresponding generating function

i≥0

A (r) i z i

The next theorem is the MacWilliams identity for the rth support weight enumerator, due

to T Kløve [6] Shiromoto [12] proved the equivalence between this result and Theorem 5, and J Simonis [13] has proved a result which is equivalent to both these results

b−1

Y

i=0

(q a − q i)

Theorem 6 [6] If A (r) (z) and B (r) (z) are the rth support weight enumerators of a linear

k-dimensional code C ⊆ F

E

q and of its dual C ⊥ for all r such that 0 ≤ r ≤ k, then the following identity holds for all m ≥ 0:

k

X

r=0

[m] r B (r) (z) = 1

q km 1 + (q m − 1)z
n

k

X

r=0



.

To generalise the rth support weight enumerators, define the rth support distribution

{A (r) E 0 | E 0 ⊆ E} of C where

v∈C 0

S(v) = E 0 } The rth support enumerator is the sum

A (r) {z e } E

E 0 ⊆E

A (r) E 0

Y

e∈E 0

z e

is given by the sum

A(0) {z e } E

+ (q − 1)A(1) {z e } E

= 1 + (q − 1)A(1) {z e } E

.

The following theorem generalises both Theorem 2 and Theorem 6 The former may be

Theorem 7 If A (r) {z e } E

and B (r) {z e } E

are the rth support enumerators of a linear k-dimensional code C ⊆ F

E

q and of its dual C ⊥ for all r such that 0 ≤ r ≤ k, then the following identity holds for all m ≥ 0:

k

X

r=0

[m] r B (r) {z e } E

=

1

q km

Y

e∈E

r=0

1 + (q m − 1)z e

o

E



.

Theorem 3 and Theorem 7 are equivalent This follows from the following oft-proved theorem (originally due to E Landberg [7])

Theorem 8 Let C be an r-dimensional subspace of F

E

q The number of ordered m-tuples

of vectors (v1, , v m) ∈ C m which span C is independent of the actual subspace C Indeed, this number equals [m] r

E

dimension r of the span of the m vectors Furthermore, Theorem 8 stipulates that there

Proposition 9 For each subset E 0 ⊆ E it holds that A [m] E 0 =

k

X

r=0

[m] r A (r) E 0 Hence,

A [m]({z e } E) =

k

X

r=0

[m] r A (r)({z e } E ).

The equivalence of Theorems 3 and 7 now follows By setting m = 1, the equivalence

of Theorems 5 and 6 is therefore also re-proved

3 The vector matroid of a linear code

E

is the matroid over E whose independent sets are the linearly independent columns of G.

corresponds to the dual code:

M ⊥

codes which are not monomially equivalent The results in this section demonstrate how some of the matroid’s properties determine many properties of the codes, in particular the various enumerators mentioned in the previous section

the sum

P (M; λ) = X

A⊆E

(−1) |A| λ r(E)−r(A)

defined by the sum

R(M; x, y) = X

A⊆E

x r(E)−r(A) y |A|−r(A)

r(E) + r ∗ (A) = |A| + r(E \ A) one may show the duality identity

The following celebrated theorem by H Crapo and G.-C Rota describes the set of supports

S(C) of a linear code We have restated the theorem slightly, in a manner similar to that

of Greene [5]

Theorem 10 [4] The m-tuple support enumerator of a linear code C ⊆F

E

q is given by

A [m] {z e } E

A⊆E

P (M C /(E \ A); q m)Y

e∈A

z e

In particular, the following corollary is obtained by setting m = 1 This result has

been derived independently in [2]

Corollary 11 The support enumerator of a linear code C ⊆F

E

q is given by

A {z e } E

A⊆E

P (M C /(E \ A); q)Y

e∈A

z e

determines the structure of the set of supports of C In turn, this implies that the

Theorems 16 and 17 below state that the m-tuple support enumerator and rth support

C Greene expresses the support weight enumerator A(z) of a code C as an evaluation

Theorem 12 [5] Let C ⊆ F

E

q be a k-dimensional linear code Then the support weight enumerator A(z) of C is given by

A(z) = (1 − z) k z n−k R

M C; qz

1− z ,

z



.

re-prove Theorem 1 This procedure is repeated by A Barg [1] who expresses rth support

Theorem 6

Theorem 13 [1] Let C ⊆F

E

q be a k-dimensional linear code If A (r) (z) is the rth support

weight enumerator of C where 0 ≤ r ≤ n, then it holds for all m ≥ 0 that

k

X

r=0

[m] r A (r) (z) = (1 − z) k z n−k R

M C; q m z

1− z ,

z



.

We will also follow this method, in order to express the rth support enumerator in

terms of matroid properties For this purpose, we will generalise the rank generating

function Let R be a domain and let R(X) be the ring of rational forms over R Associate

A⊆E

x r(E)−r(A) y |A|−r(A)Y

e∈A

g(z e) Y

f / ∈A

h(z f ).

Note that we obtain the usual rank generating function by letting g and h be the identity

closely related polynomial (a generalised Tutte polynomial for doubly weighted matroids)

Proposition 14 R g,h(M ∗ ; x, y, {z e } E ) = R h,g(M; y, x, {z e } E ).

Proof We apply the identity r(E) + r ∗ (A) = |A| + r(E \ A):

R g,h(M ∗ ; x, y, {z e } E)

A⊆E

x r ∗ (E)−r ∗ (A) y |A|−r ∗ (A)Y

e∈A

g(z e) Y

f / ∈A

h(z f)

A⊆E

x |E\A|−r(E\A) y r(E)−r(E\A)Y

e∈A

g(z e) Y

f / ∈A

h(z f)

A⊆E

y r(E)−r(A) x |A|−r(A)Y

e∈A

h(z e) Y

f / ∈A

g(z f)

= R h,g(M; y, x, {z e } E ). 

The following theorem generalises the characterisation [3] of Tutte-Groethendieck poly-nomials of a matroid due to T Brylawski A result which is closely related to the first part of Theorem 15 appears in [15]

Theorem 15 If g and h are functions in R(X), then the generalised rank generating

function R g,h is the unique function f ( M, x, y, {z e } E ) on a given minor-closed class A of matroids M and variables x ∪ y ∪ {z e } E which satisfies the following conditions:

1 f (U 0,1 , x, y, z e ) = yg(z e ) + h(z e ) and

f(U 1,1 , x, y, z e ) = g(z e ) + xh(z e );

2 If e is a loop or a coloop of M, then

f(M, x, y, {z e 0 } E ) = f ( M(e), x, y, z e )f ( M \ e, x, y, {z e 0 } E−e)

3 If e is a neither a loop nor a coloop of M, then

f(M, x, y, {z e 0 } E) =

h(z e )f ( M\e, x, y, {z e 0 } E−e ) + g(z e )f ( M/e, x, y, {z e 0 } E−e ).

Furthermore, if g(x) and h(x) are functions in R(X) such that g(x), h(x) 6= 0 0 , and

f(M, x, y, {z e } E ) is a function satisfying conditions 2 and 3, then for all e ∈ E it holds that f ( M, x, y, {z e 0 } E ) is equal to

R g,h(M; f(U 1,1 , x, y, z h(z e)− g(z e)

e) , f(U 0,1 , x, y, z e)− h(z e)

g(z e) , {z e 0 } E ).

Proof The proof is straightforward.

R g,h (U 0,1 , x, y, z e ) = x r(e)−r(∅) y |∅|−r(∅) h(z e ) + x r(e)−r(e) y |e|−r(e) g(z e)

R g,h (U 1,1 , x, y, z e ) = x r(e)−r(∅) y |∅|−r(∅) h(z e ) + x r(e)−r(e) y |e|−r(e) g(z e)

= g(z e ) + xh(z e)

that

R g,h(M, x, y, {z e 0 } E) = X

A⊆E

x r(E)−r(A) y |A|−r(A) Y

e 0 ∈A

g(z e 0) Y

f / ∈A

h(z f)

A⊆E−e

 Y

e 0 ∈A

g(z e 0) Y

f / ∈A∪e

h(z f)



F (A) ,

e )x r(M)−r(A∪e) y |A∪e|−r(A∪e) In order to

evalu-ate F (A) further, we must distinguish between three cases: e is either a loop, a coloop,

r M (A ∪ e) = r M\e (A) for all subsets A ⊆ E − e so

F (A) = yh(z e )x r(M\e)−r M\e (A) y |A|−r M\e (A)+

yg(z e )x r(M\e)−r M\e (A) y |A|−r M\e (A)

= (h(z e ) + yg(z e ))(x r(M\e)−r M\e (A) y |A|−r M\e (A) )

Since h(z e ) + yg(z e ) = R g,h (U 0,1 , x, y, z e ) = R g,h(M(e), x, y, z e), we see that

R g,h(M, x, y, {z e 0 } E ) = R g,h(M(e), x, y, z e )R g,h(M \ e, x, y, {z e 0 } E−e ).

f(M, x, y, {z e 0 } E) is a function which satisfies conditions 2 and 3 for all M ∈ A Let F0

f(U 0,1 , x, y, z e)− h(z e)

g(z e) and

f(U 1,1 , x, y, z e)− g(z e)

h(z e) ,

respectively

First note that f ( M, x, y, {z e 0 } E ) is equal to R g,h(M; F1, F0, {z e 0 } E) forM = U 0,1 , U 1,1

f(M 0 , x, y, {z e 0 } E ) is equal to R g,h(M 0 ; F

1, F0, {z e 0 } E) for the minorsM 0 =M\e 00 , M/e 00

ofM where e 00 6= e is some element of E Suppose that e 00 is a loop ofM By assumption

and by two applications of condition 2, it follows that

R g,h(M; F1, F0, {z e 0 } E)

= R g,h(M(e 00 ); F

1, F0, z e 00 )R g,h(M\e 00 ; F

1, F0, {z e 0 } E−e 00)

= f ( M(e 00 ); F

1, F0, z e 00 )f ( M\e 00 ; F

1, F0, {z e 0 } E−e 00)

= f ( M; F1, F0, {z e 0 } E )

E

shortening C/E 0 of C by the coordinate set E 0 ⊆ E is the code obtained by first removing

F

E\E 0

M C\E 0 =M C \ E 0 and M C/E 0 =M C /E 0

The following theorem generalises Theorem 12 for the m-tuple support enumerator.

Theorem 16 Let C ⊆F

E

q be a k-dimensional linear code Then

A [m] {z e } E

= R 1−x,x(M C ; q m , 1, {z e } E ).

In particular, A {z e } E

= R 1−x,x(M C ; q, 1, {z e } E ).

Proof Let M C 0 be the support matroid of each minor C 0 of C Consider the m-tuple support enumerators A [m] C 0 {z e } E 0

are well-defined since these are the only corresponding support enumerators of the

E 0

A [m] C 0 {z e } E 0

= A [m] C 0 \e 0 {z e } E 0 −e 0

= A [m] U 0,1 (z e 0 )A [m] C 0 \e 0 {z e } E 0 −e 0

contained in E 00 , then consider an m-tuple (v1, , v m ) of codewords of C 0 \ e 0 such that

∪ m

A [m] C 0 {z e } E 0

= A [m] C 0 /e 0 {z e } E 0 −e 0

+ (q m − 1)z e 0 A [m] C 0 \e 0 {z e } E 0 −e 0

A [m] C 0 \e 0 {z e } E 0 −e 0

= A [m] U

1,1 (z e 0 )A [m] C 0 \e 0 {z e } E 0 −e 0

From this, it follows that

A [m] C 0 {z e } E 0

= (1− z e 0 )A [m] C 0 /e 0 {z e } E 0 −e 0

+ z e 0 A [m] C 0 \e 0 {z e } E 0 −e 0

By induction, the identities (3.1), (3.2), (3.3), and (3.4) show that

A [m] C 0 {z e } E 0

is satisfied by the identities (3.2) and (3.3), and identity (3.4) satisfies condition 3 in

As an immediate application of Proposition 14 and Theorem 16, we may prove Theo-rem 3 as follows

B [m] {z e } E

= R 1−x,x(M C ⊥ ; q m , 1, {z e } E)

= R x,1−x(M C ; 1, q m , {z e } E) = X

A⊆E

(q m)|A|−r(A) Y

e∈A

z e Y

f / ∈A

(1− z e)

q km

X

A⊆E

(q m)r(E)−r(A)Y

e∈A

q m z e

 Y

f / ∈A

(1− z e)

q km

Y

e∈E

1 + (q m − 1)z e


R 1−x,x



M C ; q m , 1,n 1− z e

o

E



q km

Y

e∈E

1 + (q m − 1)z e


A [m]n 1− z e

o

E



The support generalisation of Theorem 13 is described in the following theorem

Theorem 17 Let C be a k-dimensional subspace of F

E

q Then for each m ≥ 0 it holds that

k

X

r=0

[m] r A (r) {z e } E

= R 1−x,x(M C ; q m , 1, {z e } E )

Proof Theorem 17 follows immediately from Proposition 9 and Theorem 16.  Furthermore, Theorem 22 follows from Theorem 16, Theorem 17, and Lemma 21 In turn, Theorem 22 implies that Theorem 12 and Theorem 13 are equivalent

Theorem 7 follows as an immediate corollary from Proposition 14 and Theorem 17

To conclude, we prove that the two latter theorems are also equivalent to Theorem 10:

R 1−x,x(M C , q m , 1, {z e } E)

A⊆E

(q m)r(E)−r(A)Y

e∈A

f / ∈A

z f

A⊆E

(q m)r(E)−r(A)X

B⊆A

e∈B

z e Y

f / ∈A

z f

A⊆E

X

B⊆A

(−1) |B| (q m)r(E)−r(A) Y

e∈B∪(E\A)

z e

A⊆E

X

B⊆A

(−1) |B| (q m

)(r(E)−r(E\A))−(r(B∪(E\A))−r(E\A)) Y

e∈A

z e

Hence,

R 1−x,x(M C , q m , 1, {z e } E) = X

A⊆E

P (M C /(E \ A); q m

e∈A

z e 

4 An alternative proof of Theorem 7

This section contains an alternative proof of Theorem 7 which does not depend on matroid theory The proof relies on Theorem 2 which, as mentioned in Section 2, follows easily from a number of results To make this section self-contained, however, a direct proof of Theorem 2 is provided

It is perhaps of interest to note that these proofs differ only very slightly, in an obvious way, from one of the two original proofs [8] of the MacWilliams identity, and from Kløve’s proof [6] of Theorem 6

Proof of Theorem 2 Let χ be a non-trivial character ofFq and define g(u) for u ∈F

E q

v∈FE q

χ hu, vi
Y

e∈S(v)

z e

u∈C g(u) in two different ways and then identify the

and B {z e } E

The first expression:

X

u∈C

g(u) = X

u∈C

X

v∈FE q

χ hu, vi
Y

e∈S(v)

z e

v∈FE q

 Y

e∈S(v)

z e X

u∈C

χ hu, vi
.

then hu, vi assumes all values ofFq an equal number of times, whence the inner sum

u∈C

g(u) = |C| X

v∈C ⊥

Y

e∈S(v)

z e=|C| · B {z e } E

For the second expression, consider g(u):

g(u) = X

v∈FE q

χ hu, vi
Y

e∈S(v)

z e

v∈FE q

Y

e∈S(v)

χ(u e v e )z e

e∈E



v e ∈Fq −0

χ(u e v e )z e



.

z e · X

a∈Fq −0

χ(a) = −z e Hence,

A [m] C 0 {z e } E 0

is satisfied by the identities. .. immediately from Proposition and Theorem 16.  Furthermore, Theorem 22 follows from Theorem 16, Theorem 17, and Lemma 21 In turn, Theorem 22 implies that Theorem 12 and Theorem 13 are...

E



.

Theorem and Theorem are equivalent This follows from the following oft-proved theorem (originally due to E Landberg [7])