Arithmetic completely regular codes

Abstract

International audience
In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.

Full text

arXiv:0911.1826v1 [math.CO] 10 Nov 2009
Arithmetic completely regular codes
J. H. Koolen∗,†W. S. Lee†W. J. Martin‡
November 10, 2009
Abstract
In this paper, we explore completely regular codes in the Ham-
ming graphs and related graphs. We focus on cartesian products of
completely regular codes and products of their corresponding coset
graphs in the additive case. Connections to the theory of distance-
regular graphs are explored and several open questions are posed.
1 Introduction
In this paper, we present the theory of completely regular codes in the Ham-
ming graph enjoying the property that the eigenvalues of the code are in
arithmetic progression. We call these arithmetic completely regular codes and
propose their full classification, modulo the classification of perfect codes of
Hamming type. Our results are strongest when the Hamming graph admits
a completely regular partition into such codes (e.g., the partition into cosets
of some additive completely regular code), since it is known that the quotient
graph obtained from any such partition is distance-regular. Using Leonard’s
Theorem, the list of possible quotients is determined, with a few special cases
left as open problems. In the case of linear arithmetic completely regular
codes, more can be said.
The techniques employed are mainly combinatorial and products of codes
as well as decompositions of codes into “reduced” codes play a fundamental
∗Pohang Mathematics Institute, POSTECH, South Korea
†Department of Mathematics, POSTECH, South Korea
‡Department of Mathematical Sciences and Department of Computer Science, Worces-
ter Polytechnic Institute, Worcester, Massachusetts, USA.
1

role. The results have impact in algebraic coding theory, but more so in
the theory of distance-regular graphs. Since, for large minimum distance,
the coset graph of any such linear code is a distance-regular graph locally
isomorphic to a Hamming graph, the analysis can be viewed as a special
case of a fairly difficult open classification problem in the theory of distance-
regular graphs. What is interesting here is how the combinatorics of the
Hamming graph gives leverage in the combinatorial analysis of its quotients,
a tool not available in the unrestricted problem.
It is important here to draw a connection between the present paper and a
companion paper, currently in preparation. In [8], we will introduce the con-
cept of Leonard completely regular codes and we will develop the basic theory
for them. We will also give several important families of Leonard completely
regular codes in the classical distance-regular graphs, demonstrating their
fundamental role in structural questions about these graphs. We conjecture
that all completely regular codes in the Hamming graphs with large enough
covering radius are in fact Leonard. The class of arithmetic codes we intro-
duce in this paper is perhaps the most important subclass of the Leonard
completely regular codes in the Hamming graphs and something similar is
likely true for the other classical families, but this investigation is left as an
open problem.
The layout of the paper is as follows. After an introductory section out-
lining the required background, we explore products of completely regular
codes in Hamming graphs. First noting (Prop. 3.1) that a completely reg-
ular product must arise from completely regular constituents, we determine
in Proposition 3.4 exactly when the product of two completely regular codes
in two Hamming graphs is completely regular. At this point, the role of the
arithmetic property becomes clear and we present a generic form for the quo-
tient matrix of such a code in Lemma 3.5. With this preparatory material
out of the way, we are then ready to present the main results in Section 3.2.
Applying the celebrated theorem of Leonard, Theorem 3.7 provides powerful
limitations on the combinatorial structure of a quotient of a Hamming graph
when the underlying completely regular partition is composed of arithmetic
codes. Stronger results are obtained in Proposition 3.9, Theorem 3.11 and
Proposition 3.13 when one makes additional assumptions about the mini-
mum distance of the codes or the specific structure of the quotient. When C
is a linear completely regular code with the arithmetic property and Chas
minimum distance at least three and covering radius at most two, we show
that Cis closely related to some Hamming code. These results are sum-
2

marized in Theorem 3.16, which gives a full classification of possible codes
and quotients in the linear case (always assuming the arithmetic and com-
pletely regular properties) and Corollary 3.17 which characterizes Hamming
quotients of Hamming graphs.
2 Preliminaries and definitions
In this section, we summarize the background material necessary to under-
stand our results. Most of what is covered here is based on Chapter 11 in
the monograph [3] by Brouwer, Cohen and Neumaier. The theory of codes
in distance-regular graphs began with Delsarte [5]. The theory of association
schemes is also introduced in the book [1] of Bannai and Ito while connec-
tions between these and related material (especially equitable partitions) can
be found in Godsil [7].
2.1 Distance-regular graphs
Suppose that Γ is a finite, undirected, connected graph with vertex set VΓ.
For vertices xand yin VΓ, let d(x, y) denote the distance between xand y,
i.e., the length of a shortest path connecting xand yin Γ. Let Ddenote the
diameter of Γ; i.e., the maximal distance between any two vertices in VΓ.
For 0 ≤i≤Dand x∈VΓ, let Γi(x) := {y∈VΓ|d(x, y) = i}and put
Γ−1(x) := ∅, ΓD+1(x) := ∅. The graph Γ is called distance-regular whenever
it is regular of valency k, and there are integers bi, ci(0 ≤i≤D) so that for
any two vertices xand yin VΓ at distance i, there are precisely cineighbors
of yin Γi−1(x) and bineighbors of yin Γi+1(x).It follows that there are
exactly ai=k−bi−cineighbors of yin Γi(x). The numbers ci,biand ai
are called the intersection numbers of Γ and we observe that c0= 0, bD= 0,
a0= 0, c1= 1 and b0=k. The array ι(Γ) := {b0, b1,...,bD−1;c1, c2,...,cD}
is called the intersection array of Γ.
From now on, assume Γ is a distance-regular graph of valency k≥2 and
diameter D≥2. Define Aito be the square matrix of size |VΓ|whose rows
and columns are indexed by VΓ with entries
(Ai)xy =(1 if d(x, y) = i
0 otherwise (0 ≤i≤D, x, y ∈VΓ).
3

We refer to Aias the ith distance matrix of Γ. We abbreviate A:= A1
and call this the adjacency matrix of Γ. Since Γ is distance-regular, we have
for 2 ≤i≤D
AAi−1=bi−2Ai−2+ai−1Ai−1+ciAi
so that Ai=pi(A) for some polynomial pi(t) of degree i. Let Abe the
Bose-Mesner algebra, the matrix algebra over Cgenerated by A. Then
dim A=D+ 1 and {Ai|0≤i≤D}is a basis for A. As Ais semi-simple
and commutative, Ahas also a basis of pairwise orthogonal idempotents
nE0=1
|VΓ|J, E1, . . . , EDo. We call these matrices the primitive idempotents
of Γ. As Ais closed under the entry-wise (or Hadamard) product ◦, there
exist real numbers qℓ
ij , called the Krein parameters, such that
Ei◦Ej=1
|VΓ|
D
X
ℓ=0
qℓ
ij Eℓ(0 ≤i, j ≤D) (1)
The graph Γ is called Q-polynomial if there exists an ordering E0,...,ED
of the primitive idempotents and there exist polynomials qiof degree isuch
that Ei=qi(E1), where the polynomial qiis applied entrywise to E1. We
recall that the distance-regular graph Γ is Q-polynomial with respect to the
ordering E0, E1, . . . , EDof its primitive idempotents provided its Krein pa-
rameters satisfy
•qℓ
ij = 0 unless |j−i| ≤ ℓ≤i+j;
•qℓ
ij 6= 0 whenever ℓ=|j−i|or ℓ=i+j≤D.
By an eigenvalue of Γ, we mean an eigenvalue of A=A1. Since Γ has
diameter D, it has at least D+ 1 eigenvalues; but since Γ is distance-regular,
it has exactly D+ 1 eigenvalues1. We denote these eigenvalues by θ0,...,θD
and, aside from the convention that θ0=k, the valency of Γ, we make no
further assumptions at this point about the eigenvalues except that they are
distinct.
2.2 Codes in distance-regular graphs
Let Γ be a distance-regular graph with distinct eigenvalues θ0=k, θ1,...,θD.
By a code in Γ, we simply mean any nonempty subset Cof VΓ. We call C
1See, for example, Lemma 11.4.1 in [7].
4

trivial if |C| ≤ 1 or C=VΓ and non-trivial otherwise. For |C|>1, the
minimum distance of C,δ(C), is defined as
δ(C) := min{d(x, y)|x, y ∈C, x 6=y}
and for any x∈VΓ the distance d(x, C) from xto Cis defined as
d(x, C) := min{d(x, y)|y∈C}.
The number
ρ(C) := max{d(x, C)|x∈VΓ}
is called the covering radius of C.
For Ca nonempty subset of VΓ and for 0 ≤i≤ρ, define
Ci={x∈VΓ|d(x, C) = i}.
Then Π(C) = {C0=C, C1,...,Cρ}is the distance partition of VΓ with
respect to code C.
A partition Π = {P0, P1,...,Pk}of VΓ is called equitable if, for all iand
j, the number of neighbors a vertex in Pihas in Pjis independent of the
choice of vertex in Pi.Following Neumaier [10], we say a code Cin Γ is
completely regular if this distance partition Π(C) is equitable2. In this case
the following quantities are well-defined:
γi=|{y∈Ci−1|d(x, y) = 1}| ,(2)
αi=|{y∈Ci|d(x, y) = 1}| ,(3)
βi=|{y∈Ci+1 |d(x, y) = 1}| (4)
where xis chosen from Ci. The numbers γi, αi, βiare called the intersection
numbers of code C. Observe that a graph Γ is distance-regular if and only
if each vertex is a completely regular code and these |VΓ|codes all have the
same intersection numbers. Set the tridiagonal matrix
U:= U(C) = 






α0β0
γ1α1β1
γ2α2β2
.........
γραρ







.
2This definition of a completely regular code is equivalent to the original definition,
due to Delsarte [5].
5

For Ca completely regular code in Γ, we say that ηis an eigenvalue of
Cif ηis an eigenvalue of the quotient matrix Udefined above. By Spec (C),
we denote the set of eigenvalues of C. Note that, since γi+αi+βi=kfor
all i,θ0=kbelongs to Spec (C).
2.3 Completely regular partitions
Given a partition Π of the vertex set of a graph Γ (into nonempty sets),
we define the quotient graph Γ/Π on the classes of Π by calling two classes
C, C′∈Π adjacent if C6=C′and Γ contains an edge joining some vertex
of Cto some vertex of C′. A partition Π of VΓ is completely regular if it is
an equitable partition of Γ and all C∈Π are completely regular codes with
the same intersection numbers. If Π = {C(1) , C(2),...,C(t)}is a completely
regular partition we write Spec (Π) = Spec (C(1) ) and we say ρis the covering
radius of Π if it is the covering radius of C(1).
Proposition 2.1 (Cf. [3, p. 352-3]) Let Π = {C(1), C (2),...,C(t)}be a
completely regular partition of any distance-regular graph Γsuch that each
C(i)has intersection numbers γi,αiand βi(0≤i≤ρ). Then Γ/Πis a
distance-regular graph with intersection array
ι(Γ/Π) = β0
γ1
,β1
γ1
,βρ−1
γ1
; 1,γ2
γ1
,...γρ
γ1,
remaining intersection numbers ai=αi−α0
γ1, and eigenvalues θj−α0
γ1for θj∈
Spec (C). All of these lie among the eigenvalues of the matrix 1
γ1(A−α0I).
Proposition 2.2 Let Πbe a non-trivial completely regular partition of a
distance-regular graph Γand assume that Spec (Π) = {η0≥η1≥ · · · ≥ ηρ}.
Then ηρ≤α0−γ1.
Proof: By Proposition 2.1, the eigenvalues of Γ/Π are η0−α0
γ1,η1−α0
γ1,...,ηρ−α0
γ1.
As Γ/Π has at least one edge, it follows that its smallest eigenvalue is at most
−1. Hence ηρ−α0
γ1≤ −1.
Question: Let Cbe a non-trivial completely regular code in a distance-
regular graph Γ. Let θ:= min{η|η∈Spec (C)}. Is it true that θ≤α0−γ1?
6

3 Codes in the Hamming graph
Let Xbe a finite abelian group. A translation distance-regular graph on X
is a distance-regular graph Γ with vertex set Xsuch that if xand yare
adjacent then x+zand y+zare adjacent for all x, y, z ∈X. A code C⊆X
is called additive if for all x, y ∈C, also x−y∈C; i.e., Cis a subgroup of X.
If Cis an additive code in a translation distance-regular graph on X, then
we obtain the usual coset partition ∆(C) := {C+x|x∈X}of X; whenever
Cis a completely regular code, it is easy to see that ∆(C) is a completely
regular partition. For any additive code Cin a translation distance-regular
graph Γ on vertex set X, the coset graph of Cin Γ is the graph with vertex
set X/C and an edge joining coset C′to coset C′′ whenever Γ has an edge
with one end in C′and the other in C′′. An important result of Brouwer,
Cohen and Neumaier [3, p353] states that every translation distance-regular
graph of diameter at least three defined on an elementary abelian group X
is necessarily a coset graph of some additive completely regular code in some
Hamming graph. Of course, the Hamming graph itself is a translation graph.
Let Qbe an abelian group with |Q|=q. Then we may identify the vertex
set of the Hamming graph H(n, q) with the group X=Qn; so the Hamming
graph can be viewed as a translation distance-regular graph in a variety of
ways. For the remainder of this paper, we will consider q-ary codes of length
nas subsets of the vertex set of the Hamming graph H(n, q). In this section
we will focus on q-ary completely regular codes, i.e. q-ary codes of length n
which are completely regular in H(n, q).
3.1 Products of completely regular codes
We now recall the cartesian product of two graphs Γ and Σ. This new graph
has vertex set VΓ×VΣ and adjacency ∼defined by (x, y)∼(u, v) precisely
when either x=uand y∼vin Σ or x∼uin Γ and y=v. Now let C
be a nonempty subset of VΓ and let C′be a nonempty subset of VΣ. The
cartesian product of Cand C′is the code defined by
C×C′:= {(c, c′)∈VΓ×VΣ|c∈Cand c′∈C′}.
We are interested in the cartesian product of codes in the Hamming graphs
H(n, q) and H(n′, q′). Note that if Cand C′are additive codes then C×C′
is also additive. In particular, if Cis a completely regular code in H(n, q)
with vertex set Qn, then Q×Cis a completely regular code in H(n+ 1, q).
7

Also if Π = {P1, . . . , Pt}is a completely regular partition of H(n, q) then
Q×Π := {Q×P1, . . . , Q×Pt}is a completely regular partition of H(n+1, q).
We say a completely regular code C⊆H(n, q) is non-reduced if C∼
=C′×Q
for some C′⊆H(n−1, q), and reduced otherwise. In similar fashion we say
that a completely regular partition is non-reduced or reduced.
The next three results will determine exactly when the cartesian product
of two arbitrary codes in two Hamming graphs is completely regular.
Proposition 3.1 Let Cand C′be non-trivial codes in the Hamming graphs
H(n, q)and H(n′, q′)respectively. If the cartesian product C×C′is com-
pletely regular in H(n, q)×H(n′, q′), then both Cand C′themselves must be
completely regular codes in their respective graphs.
Proof: If the distance partition of H(n, q) with respect to Cis {C0=
C, C1,...,Cρ}, then we easily see that every vertex (x, y) of Ci×C′is at
distance ifrom C×C′in the product graph. Moreover, the neighbors of
this vertex which lie at distance i−1 from the product code are precisely
{(u, y)|u∼x, u ∈Ci−1}; so the size of the set {u∈Qn|u∼x, u ∈Ci−1}
must be independent of the choice of x∈Ci. This shows that the intersection
numbers γiare well-defined for C. An almost identical argument gives us the
intersection numbers βi. Swapping the roles of Cand C′, we find that each
of these is a completely regular code.
Remark 3.2 Note that, if q6=q′, then the product graph is not distance-
regular. Although this case is not of primary interest, there are examples
where C×C′can still be a completely regular code (in the sense of Neumaier)
in such a graph. For instance, suppose Cis a perfect code of covering radius
one in H(n, q)and C′is a perfect code of covering radius one in H(n′, q′).
If we happen to have n(q−1) = n′(q′−1), then, by the above proposition,
C×C′is a completely regular code in H(n, q)×H(n′, q′)with intersection
numbers γ1= 1 and γ2= 2,β0= 2n(q−1) and β1=n(q−1).
It is well-known that equitable partitions are preserved under products.
If Π is an equitable partition in any graph Γ and ∆ is an equitable partition
in another graph Σ, then {P×P′|P∈Π, P ′∈∆}is an equitable partition
in the Cartesian product graph Γ ×Σ. The following special case will prove
useful in our next proposition.
8

Lemma 3.3 Consider completely regular codes Cin H(n, q)and C′in
H(n′, q′)with distance partitions Π = {C0=C,...,Cρ}and Π′={C′
0=
C′,...,C′
ρ′}respectively.
(a) The partition Ci×C′
j|0≤i≤ρ, 0≤j≤ρ′
is an equitable partition in the product graph H(n, q)×H(n′, q′).
(b) A vertex in Ci×C′
jhas all its neighbors in
(Ci−1∪Ci∪Ci+1)×C′
j∪Ci×C′
j−1∪C′
j∪C′
j+1.
(c) If Chas intersection numbers γi, αi, βiand C′has intersection numbers
γ′
i, α′
i, β′
i, then in the product graph, a vertex in Ci×C′
jhas: γineighbors
in Ci−1×C′
j;βineighbors in Ci+1 ×C′
j;γ′
jneighbors in Ci×C′
j−1;β′
j
neighbors in Ci×C′
j+1; and αi+α′
jneighbors in Ci×C′
j.
Proof: Straightforward.
Proposition 3.4 Let Cbe a non-trivial completely regular code in H(n, q)
with ρ(C) := ρ≥1and intersection numbers αi, βiand γi(0≤i≤ρ). Let
C′be a non-trivial completely regular code in H(n′, q′)with ρ(C′) := ρ′≥1
and intersection numbers α′
i, β′
iand γ′
i(0≤i≤ρ′). Assume, without loss,
that ρ≤ρ′. Then C×C′is a completely regular code in H(n, q)×H(n′, q′)
if and only if there exist integers n1and n2satisfying (a) and (b):
(a) γi=n1ifor 0≤i≤ρand γ′
i=n1ifor 0≤i≤ρ′;
(b) βρ−i=n2ifor 0≤i≤ρand β′
ρ′−i=n2ifor 0≤i≤ρ′.
In this case, C×C′has covering radius ¯ρ:= ρ+ρ′and intersection numbers
¯γi=n1iand ¯
βi=n2(¯ρ−i)for 0≤i≤¯ρ.
Proof: (⇒) Suppose first that C×C′is a completely regular code. From
Lemma 3.3, we see that C×C′has covering radius ¯ρ:= ρ′+ρ′′. For 0 ≤j≤¯ρ,
let
Sj={(x, y)|d((x, y), C ×C′) = j}.
9

Then it follows easily from the second statement in Lemma 3.3 that
Sj=[
h+i=j
Ch×C′
i.
Moreover a vertex in Ch×C′
ihas γh+γ′
ineighbors in Sj−1and βh+β′
i
neighbors in Sj+1. For j= 1, this forces γ1=γ′
1. Assume inductively that
γi=iγ1and γ′
i=iγ′
1for i < j. Then, considering a vertex in
Sj=Cr×C′
j−r∪ · · · ∪ Cj−s×C′
s
(where r= max(0, j −ρ′) and s= max(0, j −ρ)), we find
γr+γ′
j−r=γr+1 +γ′
j−r−1=···=γj−s+γ′
s.
For j≤ρ, this gives γj=jγ1=γ′
j. For ρ < j ≤ρ′, we deduce, γ′
j=jγ′
1. So
we have (a) by induction. A symmetrical argument establishes part (b).
(⇐) Considering the same partition of Sjinto cells of the form Ci×C′
j−i,
we obtain the converse in a straightforward manner: if Cand C′have inter-
section numbers given by (a) and (b), then their cartesian product C×C′is
completely regular in the product graph.
Lemma 3.5 Let k, γ, β and ρbe positive integers. The tridiagonal matrix
L=









α0ρβ
γ α1(ρ−1)β
2γ α2(ρ−2)β
.........
(ρ−1)γ αρ−1β
ργ αρ









where αi=k−iγ −(ρ−i)β(0 ≤i≤ρ), has eigenvalues Spec (L) =
{k−ti |0≤i≤ρ}where t=γ+β.
Proof: By direct verification. (See, for example, Terwilliger [12, Example
5.13].)
From now on we will look at a completely regular partition Π with
Spec (Π) = {n(q−1), n(q−1) −qt, . . . , n(q−1) −qρt}in H(n, q). As a
direct consequence of Proposition 2.2, we have the following proposition.
10

Proposition 3.6 Let Πbe a non-trivial completely regular partition of the
Hamming graph H(n, q)such that Πhas covering radius ρand Spec (Π) =
{n(q−1), n(q−1)−qt,...,n(q−1)−qρt}for some t. Then n(q−1)−α0≤qρt.
3.2 Classification
Now we will classify the possible quotients Γ/Π where Π is a completely
regular partition of Γ := H(n, q) with Spec (Π) = {n(q−1), n(q−1) −
qt, . . . , n(q−1) −qρt}for some t.
Theorem 3.7 Let Γbe the Hamming graph H(n, q)and let Πbe a completely
regular partition of its vertices. Assume that Πhas covering radius ρ≥3and
eigenvalues Spec (Π) = {n(q−1), n(q−1) −qt, . . . , n(q−1) −qρt}for some
t. Then Γ/Πis has diameter ρand is isomorphic to one of the following:
(a) a folded cube;
(b) a Hamming graph;
(c) a Doob graph (i.e., the cartesian product of some number of 4-cliques
and at least one Shrikhande graph);
(d) a distance-regular graph with intersection array {6,5,4; 1,2,6}.
Proof: Let Π be a completely regular partition with covering radius ρ≥3
and eigenvalues Spec (Π) = {n(q−1), n(q−1) −qt, . . . , n(q−1) −qρt}for
some t. Denote ηi:= k−iτ (0 ≤i≤ρ) where, k:= n(q−1)−α0
γ1and τ:= qt
γ1.
Then by Proposition 2.1, Γ/Π is a distance-regular graph with eigenvalues
{ηi|0≤i≤ρ}and diameter ρ. By [8, Proposition 3.3, Lemma 4.3], Γ/Π
is Q-polynomial with respect to E0, E1,··· , Eρwhere Eiis the primitive
idempotent corresponding to ηi.
Leonard (1982) showed that Q-polynomial distance regular-graphs with
diameter Dat least 3, fall into nine types, namely, (I), (IA), (IB), (II), (IIA),
(IIB), (IIC), (IID) and (III), where we follow the notation of Bannai and Ito
[1, p263]. Type (IID) is not possible as it only occurs if D=∞.As the
eigenvalues of Γ/Π are in arithmetic progression and ρ≥3, neither type (I),
(IA), (IB), (II), (IIA) nor (III) can occur. The only remaining possibilities are
11

types (IIB) and (IIC). Terwilliger [11] showed that a Q-polynomial distance-
regular graph Σ of type (IIB) with diameter D≥3 is either the antipodal
quotient of the (2D+ 1)-cube, or has the same intersection numbers as the
antipodal quotient of the 2D-cube. By [3, Theorem 9.2.7], it follows that
Γ/Π is either the folded graph of a n-cube for n≥7 or has intersection array
{6,5,4; 1,2,6}. For type (IIC), we have (in the notation of Bannai and Ito
[1, p270])
ηi=η0+s′i,
r1=−d−1 since D < ∞,
bi= (i+ 1 + r1)r/s∗′= (i−D)r/s∗′⇒b0=−Dr/s∗′,
ci=i(r−s′s∗′)/s∗′.
Since c1= (r−s′s∗′)/s∗′= 1,
ci=i
and
ai=k−bi−ci=−(1 + r
s∗′)·i=a1·i.
Hence Γ/Π has the same parameters as a Hamming graph. But Egawa [6]
showed that a distance-regular graph with same parameters as a Hamming
graph must be a Hamming graph or a Doob graph. Hence the theorem is
proved.
Remark 3.8 A. Theorem 3.7 also holds when Γis a Doob graph. There are
completely regular partitions of Doob graphs that are Hamming graphs
with q6= 4.For example, let s≥1be an integer. In any Doob graph
Γof diameter 4s−1
3, there exist an additive completely regular code, say
C, with covering radius 1 [9]. Let n≥1be an integer. Then Cnis
an additive completely regular code with covering radius nin Γn.Recall
∆(Cn) := {Cn+x|x∈V(Γn)}.Then Γn/∆(Cn)∼
=H(n, 4s).
B. There are 3 nonisomorphic graphs with the same intersection array, {6,5,
4; 1,2,6}, as the folded 6-cube. They are the point-block incident graphs
of the 2-(16,6,2)-designs. We do not know any example in which the
quotient graph has intersection array {6,5,4; 1,2,6}, but is not the
folded 6-cube.
12

C. If Γis a Hamming graph and Cis an additive completely regular code
in Γsatisfying the eigenvalue conditions of the above theorem, then
Γ/∆(C)is always a Hamming graph, a Doob graph or a folded cube.
We do not know any example where Γis a Hamming graph and Πis a
completely regular partition such that Γ/Πis a Doob graph. But we will
prove below in Proposition 3.13 that this cannot occur when Π = ∆(C)
for some linear completely regular code C.
D. We wonder whether it is true that if the quotient graph Γ/Πin the above
Theorem 3.7 is isomorphic to a Hamming graph or a Doob graph, then
each part in Πcan be expressed as a cartesian product of completely
regular codes with covering radius at most 2. We will show (Theorem
3.16, below) that this holds when Πis the coset partition of a linear
completely regular code satisfying the eigenvalue conditions of the above
theorem.
Proposition 3.9 Let Γbe the Hamming graph H(n, q). Let Πbe any com-
pletely regular partition of Γwhere each code Cin Πhas minimum distance
δ(C)≥2.
(a) If Γ/Π∼
=H(m, q′)then q′≥q.
(b) If q≥4then Γ/Πis not isomorphic to any Doob graph.
(c) If q≥3then Γ/Πcan not have the same intersection array as any
folded cube of diameter at least two (including the array {6,5,4; 1,2,6}).
(d) Suppose further that Π = ∆(C)for some additive code Cand Γ/Πhas
intersection array {6,5,4; 1,2,6}. Then q= 2 and Γ/Πis the folded
6-cube.
Proof: Since each Cin Π satisfies δ(C)≥2, the vertices in any clique in
Γ belong to pairwise distinct classes in Π. So the quotient Γ/Π has a clique
of size at least q, the clique number of H(n, q). This immediately implies
(a)-(c) as: H(m, q′) has no clique of size larger than q′; any Doob graph has
maximum clique size three or four; a folded cube other than K4has maximum
clique size two, as does any graph with intersection array {6,5,4; 1,2,6}.
For part (d), we clearly need only consider the case where q= 2. As
δ(C)≥2, it is easy to see that every claw K1,3in Γ/Π belongs to a unique
3-cube so that Γ/Π must be the folded 6-cube [3, Corollary 4.3.8].
13

Figure 1: Hamming graph H(2,4) and its quotient graph H(2,2)
Example 3.10 Let Γbe the Hamming graph H(2,4) and let Π := Π(Γ) =
{P1, P2, P3, P4}, where P1={00,01,10,11}, P2={02,03,12,13}, P3=
{20,21,30,31},and P4={22,23,32,33}.Then the quotient graph Γ/Πis
the Hamming H(2,2) as depicted in Figure 1. Observe that each class in
partition Πis expressible as a cartesian product:
P1={0,1} × {0,1}P2={0,1} × {2,3}
P3={2,3} × {0,1}P4={2,3} × {2,3}.
Finally note that this example can be easily extended to give H(n, q)as a
quotient of H(n, sq),n, s ≥1.
From Example 3.10, we see that the n-cube and the folded n-cube can be
quotients of H(n, 4). So Proposition 3.9 does not hold when δ(C) = 1, even
for additive codes. We next show that, if Π is the coset partition of H(n, q)
with respect to a linear code with a quotient of Hamming type, then each
class in Π may be expressed as a cartesian product of completely regular
codes with covering radius one.
Theorem 3.11 Let Γbe the Hamming graph H(n, q). Let Cbe an additive
completely regular code with minimum distance δ(C)≥2in Γand let ∆(C)be
the coset partition of Γwith respect to C. Suppose that Γ/∆(C)∼
=H(m, q′).
Then mdivides nand C=Qm
i=1 C(i), where each C(i)is a q-ary completely
regular code with covering radius 1and length n
m.
14

Proof: Let eidenote the codeword with ith position 1 and all other positions
0. Since Cis additive, the relation ≈defined by i≈jif and only if ei−ej∈C
is an equivalence relation on {1,...,n}. Let Ridenote the equivalence class
containing i. Note that
γ1=|{(j, h) : ei−hej∈C}| =|{j:∃h(ei−hej∈C)}|
since ei∈C1and δ(C)≥2. Since 0and eihave q−1 common neighbors
in Γ while Cand ei+Chave q′−1 common neighbors in Γ/∆(C), we find
that |Ri|(q−1) = γ1(q′−1). This gives |Ri|=γ1(q′−1)
q−1=n
mfor 1 ≤i≤n.
So there are exactly mequivalence classes and they all have the same size.
Let Di:= Qn|Ridenote the set of vertices aof H(n, q) satisfying ah= 0
for all h6∈ Riand let Σ be the subgraph of Γ induced by Diso Σ ∼
=H(n
m, q).
Note that Σ has the property P: if a,b∈Diand a6=bthen the common
neighbors of aand bin Γ are also in Di. Let C(i):= Di∩C.
Claim: C(i)is a q-ary completely regular code with covering radius 1 in Σ
with U(C(i)) = 0β
γ1β−γ1where β=n(q−1)
m.
Proof of Claim: We show the following two statements (i) and (ii).
(i) For any d∈Di, distance d(d, C)≤1 :
This is obvious by induction on the weight of d, as Σ has property P.
(ii) C(i)is a q-ary completely regular code of length n
m:
Assume that for d∈Di,d(d, C) = 1. Then clearly Γ(d)∩C= Γ(d)∩C(i)
by (i) and C(i):= Di∩C.
This gives us the claim.
To finish the proof of the theorem note that C(i)⊆Cso that Qm
i=1 C(i)⊆C
as Cis additive. By Proposition 3.4, Qm
i=1 C(i)is a completely regular code
with the same quotient matrix as C. So C=Qm
i=1 C(i). This shows the
theorem.
Lemma 3.12 Assume that qis a prime power. Let Cbe a non-trivial re-
duced linear completely regular code over GF (q)of length n. Then the min-
imum distance of C,δ(C), is at least 2.
Proposition 3.13 If Cis a linear q-ary completely regular code of length n,
then H(n, q)/∆(C)is never isomorphic to a Doob graph.
15

Proof: Without loss of generality, we may assume Cis reduced. Since ev-
ery Doob graph has the Shrikhande graph as a quotient, we may suppose
that H(n, q)/∆(C)∼
=Σ, the Shrikhande graph. Since Cis non-trivial and
reduced, by Lemma 3.12, it has minimum distance δ(C)≥2 and by Proposi-
tion 3.9(b), we have q≤3. The Shrikhande graph is locally a 6-gon; denote
by C1,...,C6the six vertices adjacent to vertex Cin the coset graph, with
the understanding that Ciis adjacent to Ci+1 (1 ≤i≤5) in Σ. Let eidenote
the codeword with ith position 1 and all other positions 0.
Consider first the case q= 2. With an appropriate ordering of the coor-
dinates, we may assume that e1∈C1,e1+e2∈C2and e1+e2+e3∈C. But
then e2+e3must lie in C1and a contradiction is obtained by considering
which coset contains e2: this coset must be distinct from C,C1and C2(as
δ(C)≥2) and yet adjacent to all three. But Σ contains no 4-clique.
It remains to consider q= 3. We may suppose, without loss that e1∈C1
and −e1∈C2. Now some neighbor of the zero vector — we may call it e2—
lies in C3. The word −e2must lie in some coset adjacent to both Cand C3
since δ(C)≥2. It cannot be C2since otherwise e2−e1∈Cforcing e1and
e2into the same coset. So −e2∈C4. But now we obtain a contradiction
by considering the coset containing x=−e1−e2. In Σ, this coset must
be adjacent to both C2and C4. If x∈C, then −e1and e2must belong to
the same coset, contrary to the way e2was chosen. Likewise, x∈C3is not
possible since e1and e2were chosen to belong to distinct cosets.
Lemma 3.14 Let Cbe a non-trivial, reduced, linear completely regular code
over GF (q)in the H(n, q)with parity check matrix H. Let hi(i= 1,...,n)
denote columns of Hand consider the relation ≡on {1,··· , n}given by
i≡jif hiand hjare linearly dependent. Then ≡is an equivalence relation
and its each equivalence class has size γ1:= γ1(C).
Proof: It is easy to check that the relation ≡is reflexive, symmetric and
transitive and hence an equivalence relation. As e1has γ1neighbors in C,
there are γ1−1 two-weight codewords in Cand each of these codewords is
orthogonal to every row of H. This means immediately that each equivalence
class has size γ1.
Let Cbe a non-trivial, reduced, linear completely regular code over GF (q)
in H(n, q). Let P⊆ {1,··· , n}such that each equivalence class of ≡defined
16

in Lemma 3.14 contains exactly one element from P. Let D:= C|P.Then D
is linear and has minimum distance δ(D)≥3.Moreover, if Chas Spec (C) =
{n(q−1), n(q−1) −qt, . . . , n(q−1) −qρt}and Dis completely regular, then
Dhas Spec (D) = {n(q−1)
γ1,n(q−1)−qt
γ1,...,n(q−1)−qρt
γ1}, where γ1=γ1(C).
The following result is probably known but we could not find it in the
literature.
Theorem 3.15 Let Cbe a linear completely regular code over GF (q)with
minimum distance δ(C)≥3in H(n, q).
(a) If Chas Spec (C) = {n(q−1), n(q−1) −qt}for some t≥1, then C
is a Hamming code.
(b) If Chas Spec (C) = {n(q−1), n(q−1) −qt, n(q−1) −2qt}for some
t≥1, then C=D×D, where Dis a Hamming code or q= 2 and C
is an extended Hamming code.
Proof: (a) Since Chas covering radius one and minimum distance three, it
follows immediately that Chas to be perfect.
(b) Let Gbe the graph having as vertices the codewords of C⊥in which
two codewords cand dare joined if and only if wt(c−d) = t. Then by [4,
Thm 5.7], Gis strongly regular. With µdenoting the number of common
neighbors of two nonadjacent vertices in G, we first aim to show that µ= 2
when q≥3. Let a,bbe codewords of C⊥such that wt(a) = 2t,wt(b) = t
and wt(a−b) = t. Let us consider the common neighbors of 0and a. If
c∈C⊥is a common neighbor of these two which is distinct from band a−b,
then wt(c) = t,wt(a−c) = tand wt(b−c)>0. If supp(b)∩supp(c) = ∅,
then cmust have form γ(a−b) for some γ∈GF (q) since otherwise, there
exist σ∈GF (q) for which the codeword a−b+σchas weight lying strictly
between 0 and t. But c=γ(a−b) is also impossible as d(a,c) = tand not
2t.
If 0 <|supp(b)∩supp(c)|< t, then for γ∈GF (q)− {0,1}, the codewords
b−cand b−γchave distinct weights both less than 2t. So whenever q≥3,
we must have µ= 2. In this case, Gmust be an m×m-grid graph, i.e,
H(2, m), where m2=|C⊥|and, with the same reasoning as in Theorem 3.11,
the result follows.
Suppose now that q= 2 and µ > 2. Let us consider the induced subgraph
G′of Gon the common neighbors of 0and a.Then G′is a cocktail party
graph and by Lemma 3.5, 2t≤n≤4t. For n > 2t, let aand cbe codewords
17

in C⊥such that wt(a) = wt(c) = 2t. Then as C⊥is linear, |supp(a)∩supp(c)|
is either 3
2tor t. But in both cases, we must have µ= 2,a contradiction.
For n= 2t, there exist only one codeword awith wt(a) = 2t. Since Gis
vertex transitive, Gmust be a cocktail party graph (i.e., the complement of
a perfect matching). This implies C⊥is the dual of an extended Hamming
code so Cis an extended Hamming code. Hence the result follows.
Theorem 3.16 Let Cbe a non-trivial, reduced, linear completely regular
code over GF (q). Suppose further that Chas covering radius ρin H(n, q)
and Spec (C) = {n(q−1), n(q−1) −qt, . . . , n(q−1) −qρt}for some t. Then
one of the following holds:
(a) q= 2 and
C∼
=nullsp M| · · · | M
|{z }
γ1copies
,
where M=I|1is a parity check matrix for a binary repetition
code, γ1=γ1(C)and the quotient is the folded cube;
(b) ρ= 1 and
C∼
=nullsp H| · · · | H
|{z }
γ1copies
,
where His a parity check matrix for some Hamming code and γ1=
γ1(C);
(c) ρ= 2,q= 2 and
C∼
=nullsp E| · · · | E
|{z }
γ1copies
,
where Eis a parity check matrix for a fixed extended Hamming code
and γ1=γ1(C);
(d) ρ≥2,q≥2and
C∼
=C1× · · · × C1
|{z }
ρ
,
where C1is a completely regular code with covering radius 1.
18

Proof: The reader can easily check that examples (a)–(d) are all completely
regular. (See also Bier [2].) That they are the only completely regular codes
possible follows directly from Proposition 3.9, Theorem 3.11, Proposition
3.13 and Theorem 3.15.
Note that the above result includes the following generalization of the
result of Bier [2] concerning coset graphs which are isomorphic to Hamming
graphs (cf. [3, p354]):
Corollary 3.17 Let Cbe a linear completely regular code in H(n, q)whose
coset graph is a Hamming graph H(m, q′). Then one of the following holds:
(a) ρ= 1 and
C∼
=nullsp H| · · · | H
|{z }
γ1copies
,
where His a parity check matrix for some Hamming code and γ1=
γ1(C);
(b) ρ= 2,q= 2 and
C∼
=nullsp E| · · · | E
|{z }
γ1copies
,
where Eis a parity check matrix for the extended Hamming code and
γ1=γ1(C).
(c) ρ≥2,q≥2and
C∼
=C1× · · · × C1
|{z }
ρ
,
where C1is a completely regular code with covering radius 1.
Proof: Assume Spec (C) = {θ0, θj1,...,θjm}. The eigenvalues of the Ham-
ming graph Γ/∆(C) are m(q′−1) −hq′(0 ≤h≤m) but are also obtained
from Proposition 2.1 giving
m(q′−1) −hq′=θjh−α0
γ1
(0 ≤h≤m).
This in turn gives θjh=n(q−1) −h(γ1q′) for 0 ≤h≤m. So Theorem 3.16
applies.
19

Acknowledgments
Part of this work was completed while the third author was visiting Po-
hang Institute of Science and Technology (POSTECH). WJM wishes to
thank the Department of Mathematics at POSTECH for their hospitality
and Com2MaC for financial support. JHK and LWS are partially supported
by the Basic Science Research Program through the National Research Foun-
dation of Korea (NRF) funded by the Ministry of Education, Science and
Technology (grant number 2009-0089826). JHK was also partially supported
by a grant of the Korea Research Foundation funded by the Korean Govern-
ment (MOEHRD) under grant number KRF-2007-412-J02302. WJM wishes
to thank the US National Security Agency for financial support under grant
number H98230-07-1-0025.
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21