AbstractThe Hamming space Qn is the set of binary words of length n. A partition (C1,C2,…,Cr) of Qn with quotient matrix B=[bij]r×r is equitable if for all i and j, any word in the cell Ci has exactly bij neighbors in the cell Cj. In this paper, we provide an explicit formula relating the local spectrum of cells in the face to that in the orthogonal face

Hyun, Jong Yoon

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Journal of Combinatorial Theory, Series A 119 (2012) 476–482Contents lists available at SciVerse ScienceDirect
Journal of Combinatorial Theory,
Series A
www.elsevier.com/locate/jcta
Local duality for equitable partitions of a Hamming space
Jong Yoon Hyun 1
Institute of Mathematical Science, Ewha Womans University, 11-1, Daehyun-Dong, Seodaemun-Gu, Seoul, 120-750, South Korea
a r t i c l e i n f o a b s t r a c t
Article history:
Received 25 March 2011
Available online 26 October 2011
Keywords:
Local weight enumerator
Equitable partition
Completely regular code
Eigenfunction
The Hamming space Qn is the set of binary words of length n.
A partition (C1,C2, . . . ,Cr) of Qn with quotient matrix B = [bij]r×r
is equitable if for all i and j, any word in the cell Ci has exactly
bij neighbors in the cell C j . In this paper, we provide an explicit
formula relating the local spectrum of cells in the face to that in
the orthogonal face.
© 2011 Elsevier Inc. All rights reserved.
1. Introduction
The equitable partition of a Hamming space has been extensively studied due of its importance in
coding theory [2,3,6,7] and algebraic graph theory [4]. A completely regular code of covering radius ρ
is part of the cells for an equitable (ρ + 1)-partition of a Hamming space.
In [10], A.Y. Vasil’eva proved that if C is 1-perfect (one-error-correcting perfect), then there is a
connection between the local spectrum of C with respect to a u-face and its orthogonal one. An
explicit formula describing the connection is given as well. We refer to her result as the local duality
for 1-perfect codes.
The 1-perfect codes generate an equitable bipartition of a Hamming space. It is natural to extend
the local duality for 1-perfect codes to that for arbitrary equitable partitions of a Hamming space. By
using the relations on neighboring component of the complete local spectrum, Vasil’eva also settled
the local duality for arbitrary equitable partitions of a Hamming space [11].
The purpose of the paper is to provide a more explicit formula relating the local spectrum of an
equitable partition to its orthogonal local spectrum. This permits the easy derivation of a formula
relating the local spectrum of eigenfunctions and its orthogonal local spectrum, ﬁrst obtained in [12].
E-mail address: hyun33@postech.ac.kr.
1 The results in the paper were published in part at the Workshop on Algebraic and Combinatorial Coding Theory, ACCT 2010,
Akademgorodok, Novosibirsk, Russia, Sept. 2010. The author was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MEST) (No. 2011-0010328).0097-3165/$ – see front matter © 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcta.2011.10.006
J.Y. Hyun / Journal of Combinatorial Theory, Series A 119 (2012) 476–482 477According to [12], the reconstruction problem for eigenfunctions is closely related to the local duality
for eigenfunctions.
In the next section we state our main result, Theorem 2.2, solving the local duality problem for
any equitable partition of a Hamming space in an explicit fashion. In Section 3 this result is proved,
after which some simple consequences are derived, see Corollaries 3.4, 3.6 and 3.7.
2. Main result
The Hamming space Qn is the set of binary word of length n. A subset C of Qn is called a (binary)
code of length n and the elements of C are called codewords. Let u be a word in Qn . The support of u
is the set of nonzero coordinate positions of u. Throughout this paper, we identify u with its support,
and by u¯ we denote the complement of u. The weight of u is the cardinality of its support, and it
is denoted by |u|. The Hamming distance d(u, v) between u and v equals the number of coordinate
positions in which u and v differ.
Deﬁnition 2.1. An r-partition (C1,C2, . . . ,Cr) of Qn is equitable with quotient matrix B = [bij] if for
any word v in the cell Ci , the number of neighbors in the cell C j is bij for all i, j = 1,2, . . . , r.
In the literature [2,3], the concept of equitable partitions of Qn is also referred to as “perfect
r-colorings” or “r-partition designs”.
There are known examples of equitable bipartition of Qn . A binary code C of Qn is 1-perfect if for
any word v ∈ Qn there exists a unique codeword u in C such that the number of coordinate positions
in which u and v differ is at most one. It is well known [9] that an 1-perfect code of length n exists
if and only if n = 2m − 1, m > 1. One easily veriﬁes that any 1-perfect code of length n generates
an equitable bipartition of Qn with quotient matrix
[ 0 n
1 n−1
]
. For u,w ∈ Qn , deﬁne the u-face Bwu
(centered at w) to be the set {z+ w: z ⊆ u}. If w is the zero word, then we simply denote Bwu by Bu .
By convention, if u is the zero word 0, then we set Bw0 = {w}. It should be mentioned that we do not
assume the restriction u ∩ w = ∅ for Bwu as in [5,6].
We introduce the local weight enumerator of C with respect to a u-face as follows:
Wu,wC (x) =
∑
c∈C∩Bwu
x|c+w| =
∑
c∈(C+w)∩Bu
x|c|. (2.1)
If u is the all-one word 1 in (2.1), then W 1,wC (x) becomes the weight enumerator of C with respect
to w . More generally, we deﬁne the local weight enumerator of a real valued function f of Qn . The
local weight enumerator of f with respect to a u-face is deﬁned by
Wu,wf (x) =
∑
v∈Bwu
f (v)x|v+w|.
If f is a characteristic function of a code C , then the local weight enumerator of f with respect to a
u-face is equal to that of C with the same u-face.
We are now ready to state our main result concerning the local duality for equitable partitions
of Qn .
Theorem 2.2. Let (C1,C2, . . . ,Cr) be an equitable partition of Qn with quotient matrix B. For j = 1,2, . . . , r,
let (α j1,α j2, . . . ,α jr)T be an eigenvector of B with eigenvalue λ j . Then⎡
⎢⎢⎢⎢⎣
W u¯,wC1 (x)
W u¯,wC2 (x)
...
W u¯,wCr (x)
⎤
⎥⎥⎥⎥⎦= [αi j]
−1 diag
[
Gu1(−x),Gu2(−x), . . . ,Gur (−x)
][αi j]
⎡
⎢⎢⎢⎢⎣
Wu,wC1 (−x)
Wu,wC2 (−x)
...
Wu,wCr (−x)
⎤
⎥⎥⎥⎥⎦ ,
where Gut (x) = (1− x)
n+λt
2 −|u|(1+ x) n−λt2 −|u| .
478 J.Y. Hyun / Journal of Combinatorial Theory, Series A 119 (2012) 476–4823. Proof of the main theorem
Let Vn be the 2n-dimensional vector space of real-valued functions on Qn equipped with the inner
product 〈 f , g〉 =∑x∈Qn f (x)g(x). For any u ∈ Qn , deﬁne a function f u as
f u(v) = (−1)|u∩v|.
Then the set { f u: u ∈ Qn} of characters is an orthogonal basis of Vn . If f is in Vn , then
f =
∑
u∈Qn
au f
u,
where au is the Fourier coeﬃcient at u, i.e., au = 〈 f , f u〉/〈 f u, f u〉 = 2−n〈 f , f u〉.
Let S(u) denote the sphere of radius one centered at u, i.e.,
S(u) = {v ∈ Qn: d(u, v) = 1}.
The adjacent matrix D of Qn is deﬁned by
Duv =
{
1 if d(u, v) = 1,
0 otherwise.
For f ∈ Vn , we deﬁne
(Df )(u) =
∑
v∈S(u)
f (v).
Let n be an integer and let
(1− z)x(1+ z)n−x =
∑
k0
Pk(x;n)zk.
If n is a positive integer and x is an integer with 0 x n, then
Pk(x;n) =
k∑
j=0
(−1) j
(
x
j
)(
n − x
k − j
)
which is called the Krawtchouk polynomial in x of degree k.
It is well known [9] that f u is an eigenfunction of D with eigenvalue P1(|u|;n), that is,
Df u = P1
(|u|;n) f u = (n − 2|u|) f u.
The character sum of a code C with respect to v ∈ Qn is deﬁned as
χv(C) =
∑
c∈C
(−1)|c∩v|.
The following three lemmas play an important role to prove our main result.
Lemma 3.1. (See [5].) Let C be a subset of Qn. Then
∑
c∈C
n∏
i=1
xcii =
1
2n
∑
v∈Qn
χv(C)
n∏
i=1
(1− xi)vi (1+ xi)1−vi .
Lemma 3.2. (See [6].) Let f be an eigenfunction of D with eigenvalue λ. Then the following statements are
true.
(a) (n − λ)/2 is a nonnegative integer, and hence λ is an integer.
(b) 〈 f , f u〉 = 0 if |u| = (n − λ)/2.
J.Y. Hyun / Journal of Combinatorial Theory, Series A 119 (2012) 476–482 479We denote 1C the characteristic function of a code C .
Lemma 3.3. (See [6].) Let (C1,C2, . . . ,Cr) be an equitable partition with quotient matrix B and let
(α1,α2, . . . ,αr)
T be an eigenvector of B with eigenvalue λ. Then f = ∑ri=1 αi1Ci is an eigenfunction of
D with eigenvalue λ.
We are ready to prove our main result concerning the local duality for equitable partitions of Qn .
We denote Ei the set of words of weight i.
Proof of Theorem 2.2. Recall from [2] that the quotient matrix B is diagonalizable. For j = 1,2, . . . , r,
let f j =∑ri=1 α ji1Ci , where (α j1,α j2, . . . ,α jr)T is an eigenvector of B with eigenvalue λ j . Note that
〈
f j, f
v 〉=
r∑
i=1
α ji
〈
1Ci , f
v 〉=
r∑
i=1
α jiχv(Ci). (3.1)
We claim that the following identity holds for t = 1,2, . . . , r,
r∑
j=1
αt j
∑
c∈C j
n∏
i=1
xcii =
1
2n
∑
v∈E 1
2 (n−λt )
〈
ft, f
v 〉 n∏
i=1
(1− xi)vi (1+ xi)1−vi . (3.2)
It follows from Lemma 3.1 and (3.1) that
r∑
j=1
αt j
∑
c∈C j
n∏
i=1
xcii =
1
2n
∑
v∈Qn
〈
ft, f
v 〉 n∏
i=1
(1− xi)vi (1+ xi)1−vi . (3.3)
We now apply Lemma 3.2 and Lemma 3.3 to obtain our claim (3.2).
Let u be a word in Qn . Setting xi = 0 for all i ∈ u¯ in (3.2), we have
r∑
j=1
αt j
∑
c∈C j∩Bu
n∏
i=1
xcii =
1
2n
∑
v∈E 1
2 (n−λt )
〈
ft, f
v 〉 n∏
i=1
(1− xi)vi (1+ xi)1−vi
∣∣∣∣
xi=0,i∈u¯
. (3.4)
Setting x = xi for all i ∈ u in (3.4), we have
r∑
j=1
αt j
∑
c∈C j∩Bu
x|c| = 1
2n
∑
v∈E 1
2 (n−λt )
〈
ft, f
v 〉(1− x)|u∩v|(1+ x)|u|−|u∩v|, (3.5)
for all t = 1,2, . . . , r. Put
F ut (x) =
1
2n
∑
v∈E 1
2 (n−λt )
〈
ft, f
v 〉(1− x)|u∩v|(1+ x)|u|−|u∩v| (t = 1,2, . . . , r).
Then the system of r equations in (3.5) becomes
[αi j]
⎡
⎢⎢⎢⎢⎣
Wu,0C1 (x)
Wu,0C2 (x)
...
u,0
⎤
⎥⎥⎥⎥⎦=
⎡
⎢⎢⎢⎣
F u1 (x)
F u2 (x)
...
F u(x)
⎤
⎥⎥⎥⎦ . (3.6)WCr (x) r
480 J.Y. Hyun / Journal of Combinatorial Theory, Series A 119 (2012) 476–482Notice that
F u¯t (−x) =
1
2n
∑
v∈E 1
2 (n−λt )
〈
ft, f
v 〉(1+ x)|u¯∩v|(1− x)|u¯|−|u¯∩v|
= (1− x) 12 (n+λt )−|u|(1+ x) 12 (n−λt )−|u|F ut (x). (3.7)
Put
Gut (x) = (1− x)
1
2 (n+λt )−|u|(1+ x) 12 (n−λt )−|u|.
After replacing x (resp. u) by −x (resp. u¯) in (3.6), apply (3.7) to obtain
[αi j]
⎡
⎢⎢⎢⎢⎣
W u¯,0C1 (−x)
W u¯,0C2 (−x)
...
W u¯,0Cr (−x)
⎤
⎥⎥⎥⎥⎦= diag
[
Gu1(x),G
u
2(x), . . . ,G
u
r (x)
]
⎡
⎢⎢⎢⎣
F u1 (x)
F u2 (x)
...
F ur (x)
⎤
⎥⎥⎥⎦ . (3.8)
Using the fact that B is diagonalizable and combining (3.6) and (3.8), we have
⎡
⎢⎢⎢⎢⎣
W u¯,0C1 (x)
W u¯,0C2 (x)
...
W u¯,0Cr (x)
⎤
⎥⎥⎥⎥⎦= [αi j]
−1 diag
[
Gu1(−x),Gu2(−x), . . . ,Gur (−x)
][αi j]
⎡
⎢⎢⎢⎢⎣
Wu,0C1 (−x)
Wu,0C2 (−x)
...
Wu,0Cr (−x)
⎤
⎥⎥⎥⎥⎦ . (3.9)
We remark that if (C1,C2, . . . ,Cr) is an equitable partition of Qn with quotient matrix B , then (C1 +
w,C2 + w, . . . ,Cr + w) is also the equitable partition of Qn with the same quotient matrix. It then
follows from (2.1) and (3.9) that
⎡
⎢⎢⎢⎢⎣
W u¯,wC1 (x)
W u¯,wC2 (x)
...
W u¯,wCr (x)
⎤
⎥⎥⎥⎥⎦= [αi j]
−1 diag
[
Gu1(−x),Gu2(−x), . . . ,Gur (−x)
][αi j]
⎡
⎢⎢⎢⎢⎣
Wu,wC1 (−x)
Wu,wC2 (−x)
...
Wu,wCr (−x)
⎤
⎥⎥⎥⎥⎦ .
This proves Theorem 2.2. 
Setting u = 0 in Theorem 2.2, leads to the following corollary.
Corollary 3.4. Let (C1,C2, . . . ,Cr) be an equitable partition of Qn with quotient matrix B. For j = 1,2, . . . , r,
let (α j1,α j2, . . . ,α jr)T be an eigenvector of B with eigenvalue λ j . If w ∈ Cl , then for t = 1,2, . . . , r, we have
W 1,wCt (x) =
r∑
i=1
αilβti(1− x) 12 (n−λi)(1+ x) 12 (n+λi),
where βi j is an (i, j) entry of [αi j]−1 .
We remark that the formula obtained from Corollary 3.4 is more effective computationally than
the matrix formula given in [8].
Example 3.5. Let n = 2m − 3,m > 2. D.S. Krotov proved in [7] that an (n,2n−m,3) code C1 generates
an equitable partition (C1,C2,C3,C4) of Qn with quotient matrix
J.Y. Hyun / Journal of Combinatorial Theory, Series A 119 (2012) 476–482 481B =
⎡
⎢⎣
0 1 n − 1 0
1 0 n − 1 0
1 1 n − 4 2
0 0 n − 1 1
⎤
⎥⎦ .
The set of eigenvalues of B is {−3,−1,1,n}. Let (D1, D2, D3, D4) be any equitable partition of Qn
with the same quotient matrix B . It follows from Corollary 3.4 that if w is in D2, then W
1,w
D2
(x)
becomes
n − 1
4n + 12 (1− x)
n+3
2 (1+ x) n−32 + 1
2
(1− x) n+12 (1+ x) n−12
+ 1
4
(1− x) n−12 (1+ x) n+12 + 1n+3 (1+ x)n.
In [1], M.R. Best and A.E. Brouwer used the weight enumerator of D2 in another fashion to show that
the maximum cardinality of length n = 2m − 3 and minimum distance 3 equals 2n−m .
An equitable bipartition of Qn is equivalent to a completely regular code of covering radius one,
and the quotient matrix of an equitable bipartition has two distinct eigenvalues, see [2] for details.
The following corollary is a generalization of the local duality for 1-perfect codes established in [10].
Corollary 3.6. Let (C1,C2) be an equitable bipartition of Qn with quotient matrix
[ i j
k l
]
. Then
W u¯,wC1 (x) =
k
j + k (1+ x)
|u¯|
+ (1− x) 12 ( j+k)−|u|(1+ x) 12 (i+l)−|u|
(
− k
j + k (1− x)
|u| + Wu,wC1 (−x)
)
.
In particular, if w ∈ C1 , then
W 1,wC1 (x) =
k
j + k (1+ x)
n + j
j + k (1− x)
1
2 ( j+k)(1+ x) 12 (i+l).
Proof. It is a straightforward consequence of the fact that (− j,k) (resp. (1,1)) is an eigenvector of
the quotient matrix with eigenvalue i − k (resp. n). 
Next, we derive directly, without the use of induction, the local duality for eigenfunctions of D
established in [12]. Corollary 3.7 plays an important role in settling the reconstruction problem for
eigenfunctions of D [12].
Corollary 3.7. (See [12].) Let f be an eigenfunction of D with eigenvalue λ. Then
W u¯,wf (x) = (1+ x)
1
2 (n+λ)−|u|(1− x) 12 (n−λ)−|u|Wu,wf (−x).
In particular,
W 1,wf (x) = f (w)(1+ x)
1
2 (n+λ)(1− x) 12 (n−λ).
Proof. The partition ({v})v∈Qn is equitable with quotient matrix D . We expand f in the standard
basis {1{v}: v ∈ Qn} as follows:
f =
∑
v∈Qn
f (v)1{v}.
Since f is an eigenfunction of D with eigenvalue λ, we can apply (3.5) to obtain
482 J.Y. Hyun / Journal of Combinatorial Theory, Series A 119 (2012) 476–482∑
v∈Qn
f (v)
∑
c∈{v}∩Bu
x|c| = 1
2n
∑
v∈E 1
2 (n−λ)
〈
f , f v
〉
(1− x)|u∩v|(1+ x)|u|−|u∩v|. (3.10)
The left-hand side of (3.10) becomes the local weight enumerator of f with respect to the zero word.
It then follows from (3.7) that
W u¯,0f (x) = (1+ x)
1
2 (n+λ)−|u|(1− x) 12 (n−λ)−|u|Wu,0f (−x). (3.11)
We deﬁne a function g by g(u) = f (u + w) that is also an eigenfunction of D with eigenvalue λ. To
obtain the result, we apply g to (3.11). 
Acknowledgments
The author would like to thank the anonymous referees for their helpful comments improving the
clarity of this paper. The author also would like to thank Hoon Hong and Jack H. Koolen for their
helpful comments for this paper.
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Local duality for equitable partitions of a Hamming space 

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Local duality for equitable partitions of a Hamming space

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