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Partition Designs with two subsets of the Hamming Space
Partition Designs with two subsets of the Hamming Space Roberto Canogar Departamento de Matemticas Facultad de Ciencias, U.N.E.D. Madrid 28040, Spain [email protected]
Abstract A partition design with two sets in the binary Hamming graph has associated its (2 2 positive integer) adjacency matrix. Weconsider the problem of nding all a b such matrices. We will prove that the matrix is of this type if and only if c d
(i)
d
=
a
+b
c
, (ii)
b+c
gcd(b;c)
is a power of two; and (iii)
a
(
), where
f b; c
f
is
an unknown function but with the following bounds: 21 (b c) + 14 + c(b c) f (b; c) c gcd(b; c) for b c. We will also treat the problem of determining if a matrix is the adjacency matrix of a unique (up to isometry) partition design of the binary Hamming graph. Key words:
Hamming graph, partition designs, Fourier transform
1 Introduction Let F = f0; 1g be the eld with two elements. We will denote by F the m-dimensional vector space over F. The binary Hamming graph, H (m; 2), has as vertices the elements of F . The edges join vertices at distance one, where the distance between two vertices is the number of di erent coordinates. m
m
De nition 1 Given a graph G = (V; E ), a partition design of G with adja-
cency matrix M = (m )1 is de ned as a partition Y1 ; : : : ; Y of the vertex set V such that for any x 2 Y we have that #fy 2 Y j(x; y ) 2 E g = m for all 1 i; j r. It is important to note that the number m does not depend on the vertex x of Y . We say that two partition designs are equivalent if there is an isometry of the graph that maps one partition into the other one. ij
i;j
r
r
i
j
ij
ij
i
Preprint submitted to Elsevier Preprint
12 May 2000
In the literature we nd several terms which are equivalent to the term partition design. Partition designs can be found in Camion, Corteau & Delsarte [2]. But probably, the term equitable partitions is more popular and can be found in the papers by Godsil (see for example [4]). But also the term regular partitions is in use (see Appendix A.4 in [1]). The case of partition designs in the Hamming graph has been treated in detail in [2]. The study of partition designs is motivated by the fact that they produce orthogonal arrays and completely regular codes (see de nitions in MacWilliams & Sloane [5]). In [3] we prove that if M is the adjacency matrix of a partition design and 1 2 r are its eigenvalues then any set of the partition is an orthogonal array of strength 1 2 2 1. In this work we will study the partition designs of the Hamming graph that have only two sets. Note that if
a b
is the adjacency matrix of a partition
cd design of H (m; 2) then d = a + b c and m = a + b. Without loss of generality we may assume that b c. So it makes sense to de ne (a; b; c) as a binary b a Hamming triple if b c and is the adjacency matrix of some c a+b c partition design of H (a + b; 2). The aim of this work will be to nd the set of all binary Hamming triples. We will almost obtain a complete answer to this problem. From now on we will assume that all partition designs are of binary Hamming graphs.
2
Main Result
There is a strong restriction on the parameters b and c in a binary Hamming triple (a; b; c).
Proposition 2
If
(a; b; c)
is a binary Hamming triple then
of two.
b+c gcd(b;c)
is a power
If M is a matrix with natural coeÆcients and equal rows that add up to a power of two then M I is the adjacency matrix of a partition design. Moreover, given the adjacency matrix M of a partition design we will see that M + I and tM , for t integer greater than one, correspond to adjacency matrices of new partition designs. Gluing the previous results together we can state our main theorem: 2
Theorem 3 Let us de ne the set D
:=
f(
and the function f (b; c)
f
b; c)
:
D
:= min
D
2 j N
!
a
N b
Z
2
by c
and
b
+c
gcd(b; c)
is a power of twog
+ by
a + Z c
b a
+b
c
is the adjacency
matrix
of a partition design
:
Then the set of binary Hamming triples is given by
f( Moreover, the function 1 2
f
a; b; c)
c)
Z
+ Dja f (b; c)g:
is bounded by
(b
2
+
1 4
+ c (b
c)
f (b; c)
c
gcd(b; c):
The cases where the binary Hamming triple (a; b; c) corresponds to a unique partition design are the most interesting. In the following result we show that a large subset of all binary Hamming triples do not have this property.
Theorem 4 Let (a; b; c) be a binary Hamming triple. Then for t 2 there are more than one partition design with matrix (ta; tb; tc) except in the case that t = 2 and (b; c) = (3; 1), and the case that a = 0 and b = c. References [1]
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance Regular Graphs. Springer (Berlin) 1989.
[2]
P. Camion, B. Courteau, P. Delsarte, On r-partition designs in Hamming spaces.
Applicable Algebra in Engineering, Communication and Computing
2, 147-162 [3]
R.
(1992)
Canogar,
Reconstructing
partition
designs
in
association
schemes,
submitted for publication. [4]
C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
[5]
F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland Publ. Co., Amsterdam, 1977.
3
Abstract A partition design with two sets in the binary Hamming graph has associated its (2 2 positive integer) adjacency matrix. Weconsider the problem of nding all a b such matrices. We will prove that the matrix is of this type if and only if c d
(i)
d
=
a
+b
c
, (ii)
b+c
gcd(b;c)
is a power of two; and (iii)
a
(
), where
f b; c
f
is
an unknown function but with the following bounds: 21 (b c) + 14 + c(b c) f (b; c) c gcd(b; c) for b c. We will also treat the problem of determining if a matrix is the adjacency matrix of a unique (up to isometry) partition design of the binary Hamming graph. Key words:
Hamming graph, partition designs, Fourier transform
1 Introduction Let F = f0; 1g be the eld with two elements. We will denote by F the m-dimensional vector space over F. The binary Hamming graph, H (m; 2), has as vertices the elements of F . The edges join vertices at distance one, where the distance between two vertices is the number of di erent coordinates. m
m
De nition 1 Given a graph G = (V; E ), a partition design of G with adja-
cency matrix M = (m )1 is de ned as a partition Y1 ; : : : ; Y of the vertex set V such that for any x 2 Y we have that #fy 2 Y j(x; y ) 2 E g = m for all 1 i; j r. It is important to note that the number m does not depend on the vertex x of Y . We say that two partition designs are equivalent if there is an isometry of the graph that maps one partition into the other one. ij
i;j
r
r
i
j
ij
ij
i
Preprint submitted to Elsevier Preprint
12 May 2000
In the literature we nd several terms which are equivalent to the term partition design. Partition designs can be found in Camion, Corteau & Delsarte [2]. But probably, the term equitable partitions is more popular and can be found in the papers by Godsil (see for example [4]). But also the term regular partitions is in use (see Appendix A.4 in [1]). The case of partition designs in the Hamming graph has been treated in detail in [2]. The study of partition designs is motivated by the fact that they produce orthogonal arrays and completely regular codes (see de nitions in MacWilliams & Sloane [5]). In [3] we prove that if M is the adjacency matrix of a partition design and 1 2 r are its eigenvalues then any set of the partition is an orthogonal array of strength 1 2 2 1. In this work we will study the partition designs of the Hamming graph that have only two sets. Note that if
a b
is the adjacency matrix of a partition
cd design of H (m; 2) then d = a + b c and m = a + b. Without loss of generality we may assume that b c. So it makes sense to de ne (a; b; c) as a binary b a Hamming triple if b c and is the adjacency matrix of some c a+b c partition design of H (a + b; 2). The aim of this work will be to nd the set of all binary Hamming triples. We will almost obtain a complete answer to this problem. From now on we will assume that all partition designs are of binary Hamming graphs.
2
Main Result
There is a strong restriction on the parameters b and c in a binary Hamming triple (a; b; c).
Proposition 2
If
(a; b; c)
is a binary Hamming triple then
of two.
b+c gcd(b;c)
is a power
If M is a matrix with natural coeÆcients and equal rows that add up to a power of two then M I is the adjacency matrix of a partition design. Moreover, given the adjacency matrix M of a partition design we will see that M + I and tM , for t integer greater than one, correspond to adjacency matrices of new partition designs. Gluing the previous results together we can state our main theorem: 2
Theorem 3 Let us de ne the set D
:=
f(
and the function f (b; c)
f
b; c)
:
D
:= min
D
2 j N
!
a
N b
Z
2
by c
and
b
+c
gcd(b; c)
is a power of twog
+ by
a + Z c
b a
+b
c
is the adjacency
matrix
of a partition design
:
Then the set of binary Hamming triples is given by
f( Moreover, the function 1 2
f
a; b; c)
c)
Z
+ Dja f (b; c)g:
is bounded by
(b
2
+
1 4
+ c (b
c)
f (b; c)
c
gcd(b; c):
The cases where the binary Hamming triple (a; b; c) corresponds to a unique partition design are the most interesting. In the following result we show that a large subset of all binary Hamming triples do not have this property.
Theorem 4 Let (a; b; c) be a binary Hamming triple. Then for t 2 there are more than one partition design with matrix (ta; tb; tc) except in the case that t = 2 and (b; c) = (3; 1), and the case that a = 0 and b = c. References [1]
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance Regular Graphs. Springer (Berlin) 1989.
[2]
P. Camion, B. Courteau, P. Delsarte, On r-partition designs in Hamming spaces.
Applicable Algebra in Engineering, Communication and Computing
2, 147-162 [3]
R.
(1992)
Canogar,
Reconstructing
partition
designs
in
association
schemes,
submitted for publication. [4]
C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
[5]
F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland Publ. Co., Amsterdam, 1977.
3