Decompositions of Edge-Colored Digraphs: A New Technique in the Construction of Constant-Weight Codes and Related Families

0.37 MB·

by  Yeow Meng Chee

Decompositions of Edge-Colored Digraphs: A New
Technique in the Construction of Constant-Weight
Codes and Related Families
Yeow Meng Chee∗, Fei Gao†, Han Mao Kiah∗, Alan Chi Hung Ling‡, Hui Zhang∗, Xiande Zhang∗
∗
School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore,
emails: [email protected], [email protected], [email protected], [email protected]
†
Institute of High Performance Computing, Agency for Science, Technology and Research, Singapore, email: [email protected]
‡
Department of Computer Science, University of Vermont, USA, email: [email protected]
Abstract—We demonstrate that certain Johnson-type bounds II. PRELIMINARY
areasymptoticallyexactforavarietyofclassesofcodes,namely, TheringZ/qZisdenotedbyZ .Forpositiveintegern,the
4 constant-composition codes, nonbinary constant-weight codes q
1 and multiply constant-weight codes. This was achieved via an set {1,2,...,n} is denoted by [n].
0 interesting application of the theory of decomposition of edge- Let ZXq denoted the set of vectors whose elements belong
2 colored digraphs. to Z and are indexed by X. A q-ary code of length n is
q
n Key words and phrases: Johnson-type bounds, constant- then a set C ⊆ ZX with |X| = n. The support of a vector
q
a composition codes, constant-weight codes, multiply constant- u ∈ ZX, denoted supp(u), is the set {x ∈ X : u (cid:54)= 0}. The
J weight codes Hammqing weight of u ∈ ZX is defined as ||u|| =x |supp(u)|.
9 I. INTRODUCTION q
The distance induced by this norm is called the Hamming
1 In 1970s, Wilson [1]–[4] demonstrated that the elementary distance, so that d(u,v) = ||u−v||, for u,v ∈ ZX. A code
q
] necessary conditions for the existence of balanced incomplete C is said to have distance d if d(u,v) ≥ d for all distinct
O blockdesignsareasymptoticallysufficient.Hisworktherefore u,v ∈ C. The composition of a vector u ∈ ZX is the tuple
q
C determinedthesizeofanoptimalbinaryconstant-weightcode w = [w ,...,w ], where w = |{x ∈ X : u = i}|, where
1 q−1 i x
h. of weight w and distance 2w−2, provided that the length n i∈Zq\{0}. Unless mentioned otherwise, we always assume
is sufficiently large and satisfies certain congruence classes. w ≥w ≥···≥w .
t 1 2 q−1
a Moreover, these optimal codes meet the Johnson bound and
m his results show that Johnson bound is asymptotically exact A. Constant-Weight Codes and Constant-Composition Codes
[ for fixed weight w and distance 2w−2. A code C is said to have constant-weight w if every

Since Wilson’s seminal work, there have been interesting codeword in C has weight w, and has constant-composition
2
developmentsinbothareasofcombinatorialdesignandcoding w if every codeword in C has composition w. We refer to
v
5 theory.Intheformer,Wilson’sideasmaturedintoLamkenand a q-ary code of length n, distance d, and constant-weight w
2 Wilson’stheoryofdecompositionofedge-coloreddigraphs[5] as a CWC(n,d,w) . If in addition, the code has constant-
q
9 and the theory has been used extensively in establishing the composition w, then it is referred to as a CCC(n,d,w) . The
q
3
asymptoticexistenceofmanyclassesofcombinatorialdesigns. maximum size of a CWC(n,d,w) is denoted A (n,d,w)
. q q
1 Theoretical developments also extend Lamken and Wilson’s while the maximum size of a CCC(n,d,w) is denoted
q
0 results to other classes of decompositions. Of particular inter- A (n,d,w). Any CWC(n,d,w) or CCC(n,d,w) attaining
4 q q q
est is the class of superpure decompositions [6]. the maximum size is called optimal.
1
v: beeOnngetnheeraolitzheedrtohacnodn,stabnint-acroymcpoonsisttiaonnt-cwoediegshatndcondoensbihnaavrye CCCCo(nns,idde,rwa) coismaplsoositaioCnWwCa(nnd,dl,ewt)w.= (cid:80)qi=−11wi. Then a
i q q
X constant-weight codes and have found applications in power- Johnson-type bounds for constant-weight codes and
r line communications [7], [8], frequency hopping [9], coding constant-composition codes have been derived.
a for bandwidth-limited channels [10]. More recently, multiply
Lemma 1 (Svanstro¨m [12], Svanstro¨m et al. [13]).
constant-weight codes are introduced in an application for
(cid:22) (cid:23)
physicallyunclonablefunctions[11].Notsurprisingly,inthese n
A (n,d,w)≤ A (n−1,d,[w −1,...,w ]) ,
generalizations,Johnson-typeupperboundshavebeenderived. q w q 1 q−1
1
Therefore, a natural question is whether the advanced tech- (cid:22)(q−1)n (cid:23)
A (n,d,w)≤ A (n−1,d,w−1) .
niques of decompositions of edge-colored digraphs are rele- q w q
vant in constructing optimal codes and whether Johnson-type
bounds are asymptotically exact for these generalizations. We
answer both questions in the affirmative for certain distances Apply the fact that A (n,2w,w) = (cid:98)n/w(cid:99) and
q
where the weight is fixed. A (n,2w,w) = (cid:98)n/w(cid:99) (see Fu et al. [14], Chee et al. [15])
q
to Lemma 1, we have the following upper bounds: K if each edge-colored digraph in F is isomorphic to some
(cid:22) (cid:22) (cid:23)(cid:23) G ∈ G. Furthermore, a G-decomposition of K is said to be
n n−1
A (n,2w−2,w)≤ , (1) superpure if any two distinct edge-colored subgraphs in F
q w w−1
1 share at most two vertices.
(cid:106) (cid:106) (cid:107)(cid:107)
n n−1 , if w >w , Lamken and Wilson [5] exhibited the asymptotic existence
Aq(n,2w−3,w)≤(cid:106)w1 (cid:106)w1−1(cid:107)(cid:107) 1 2 (2) of decompositions of Kn(r) for a fixed family of digraphs.
 n n−1 , otherwise, Hartmann [6] later extended their results to superpure decom-
w1 w1
(cid:22)(q−1)n(cid:22)n−1(cid:23)(cid:23) positions. To state the theorems, we require more concepts.
Aq(n,2w−2,w)≤ w w−1 , (3) Consider an edge-colored digraph G = (V,C,E) with
(cid:22) (cid:22) (cid:23)(cid:23) |C| = r. Let ((u,v),c) ∈ E denote a directed edge from
(q−1)n (q−1)(n−1)
A (n,2w−3,w)≤ . (4) u to v, colored by c. For any vertex u and color c, define
q w w−1
the indegree and outdegree of u with respect to c, to be the
In this paper, we show that the above inequalities are exact number of directed edges of color c entering and leaving u
provided that n is sufficiently large and n satisfies certain respectively. Then for vertex u, we define the degree vector
congruence conditions. To do so, we apply the theory of of u in G, denoted by τ(u,G), to be the vector of length 2r,
decompositions of edge-colored digraphs to construct optimal τ(u,G) (cid:44) (in (u,G),out (u,G),...,in (u,G),out (u,G)).
1 1 r r
codes meeting the above bounds. Defineα(G)tobethegreatestcommondivisoroftheintegers
We list previous similar asymptotic or exact results. t such that the 2r-vector (t,t,...,t) is a nonnegative integral
(i) Results for A (n,d,w) are known linear combination of the degree vectors τ(u,G) as u ranges
q
over all vertices of all digraphs G∈G.
(a) for all w and d=2w−1 [16];
(b) for all d where w≤3 [16], [17]; For each G = (V,C,E) ∈ G, let µ(G) be the edge vector
(c) for (q,d,w)=(3,5,4) [18]. of length r given by µ(G) (cid:44) (m (G),m (G),...,m (G))
1 2 r
(ii) Results for Aq(n,d,w) are known where mi(G) is the number of edges with color i in G. We
(a) for all w and d=2w−1 [16], [19]; denote by β(G) the greatest common divisor of the integers
(b) for all q and (d,w)∈{(3,2),(4,3),(5,3)} [19]–[21]; m such that (m,m,...,m) is a nonnegative integral linear
(c) for (q,d,w)∈{(3,5,4),(3,6,4),(4,5,4)} [22]–[24].
combination of the vectors µ(G), G ∈ G. Then G is said to
B. Multiply Constant-Weight Codes be admissible if (1,1,...,1) can be expressed as a positive
rational combination of the vectors µ(G), G∈G.
Consider a binary code C ⊆ Z[m]×[n] of constant-weight
2 Below is the main theorem we allude to in our proofs.
mw and distance d. The code C is said to be of multiply
constant-weight w if for u ∈ C, i ∈ [m], the subword Theorem3(LamkenandWilson[5],Hartmann[6]). LetG be
(u ) is of constant-weight w. Denote such a code an admissible family of edge-colored digraphs with r colors.
i,j j∈[n]
by MCWC(m,n,d,w). Similarly, the maximum size of a Then there exists a constant n0 (resp. n1) such that a (resp.
MCWC(m,n,d,w) is given by M(m,n,d,w) and a multiply superpure) G-decomposition of Kn(r) exists for every n ≥ n0
constant-weightcodeattainingthemaximumsizeissaidtobe (resp. n ≥ n1) satisfying: n(n − 1) ≡ 0 (mod β(G)) and
optimal. A Johnson-type bound can again be derived. n−1≡0 (mod α(G)).
Lemma 2 (Chee et al. [11]). WeapplyTheorem3toconstructcodesthatmeettheupper
(cid:106)n (cid:106)n(cid:107)(cid:107) bounds (1)–(5). To do so, we have two main steps.
M(m,n,2mw−2,w)≤ . (5) (A) Define a family G of edge-colored digraphs on a set of
w w
r colors. We then show that a G-decomposition of K(r)
Again,weverify(5)isexactprovidednissufficientlylarge n
resultsinacodeoflengthnsatisfyingcertainweightand
and satisfies certain congruence conditions. Next, we describe
distance properties.
the main tool in our constructions of optimal codes.
(B) Compute α(G) and β(G) and hence determine the con-
C. Decomposition of Edge-Colored Complete Digraphs gruences classes that n needs to satisfy.
In this paper, we focus on step (A), describing the con-
Denote the set of all ordered pairs of a finite set X with
distinct components by (cid:0)X(cid:1). An edge-colored digraph is a struction of G and establishing the correspondence to certain
2 codes. As the computations in step (B) are usually tedious
triple G=(V,C,E), where V is a finite set of vertices, C is
a finite set of colors and E is a subset of (cid:0)V(cid:1)×C. Members and analogous, we exhibit one computation for illustrative
2 purposes and defer other computations to the appendices.
of E are called edges. The complete edge-colored digraph on
n vertices with r colors, denoted by K(r), is the edge-colored III. ASYMPTOTICALLYEXACTJOHNSON-TYPEBOUNDS
n
digraph(V,C,E),where|V|=n,|C|=r andE =(cid:0)V(cid:1)×C. FORCONSTANT-COMPOSITIONCODES
2
A family F of edge-colored subgraphs of an edge-colored Fix w = [w ,w ,...,w ] with w ≥ ··· ≥ w > 0
1 2 q−1 1 q−1
digraphK isadecompositionofK ifeveryedgeofK belongs and let w = (cid:80)q−1w . In this section, we construct infinite
i=1 i
to exactly one member of F. Given a family of edge-colored families of optimal CCC(n,d,w) for d = 2w −2 or d =
q
digraphs G, a decomposition F of K is a G-decomposition of 2w−3, and establish the asymptotic exactness of (1) and (2).
A. When Distance d=2w−2 Example 2. Let w =[2,1]. Consider the following superpure
G(w)-decomposition of K(2):
Consider the following characterization of codes of 5
constant-weight w with distance 2w − 2. Note that since 2(cid:84)(cid:84) (cid:111)(cid:111) (cid:47)(cid:47)(cid:53)(cid:53)4 3(cid:84)(cid:84) (cid:111)(cid:111) (cid:47)(cid:47)(cid:53)(cid:53)5 1(cid:84)(cid:84) (cid:111)(cid:111) (cid:47)(cid:47)(cid:53)(cid:53)3
theconstant-compositioncodesareconstant-weightcodes,the (cid:21)(cid:21) (cid:21)(cid:21) (cid:21)(cid:21)
(cid:118)(cid:118) (cid:118)(cid:118) (cid:118)(cid:118)
lemma is applicable for both classes of codes. 3 2 4
Lemma 4. The following are necessary and sufficient for a 1(cid:84)(cid:84) (cid:111)(cid:111) (cid:47)(cid:47)(cid:53)(cid:53)2 4(cid:84)(cid:84) (cid:111)(cid:111) (cid:47)(cid:47)(cid:53)(cid:53)5
code C of constant-weight w to have distance 2w−2: (cid:21)(cid:21) (cid:21)(cid:21)
(cid:118)(cid:118) (cid:118)(cid:118)
(C1) For i ∈ [q −1], the ordered pairs in the set {(x,y) : 5 1
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
u =i,y ∈supp(u)\{x},u∈C} are distinct. 1 2 2 1 3 1 4 2 5 3
x
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
(C2) For any u,v∈C, |supp(u)∩supp(v)|≤2. 1 3 2 4 3 5 4 5 5 4.
The corresponding code is then given by {(0,1,2,1,0),
Definition of the family G(w). For fixed w, define an edge-
(0,2,1,0,1),(1,0,1,2,0),(1,1,0,0,2),(2,0,0,1,1)}, which
colored digraph G(w)=(V[w],C[w],E[w]), where
is indeed a CCC(5,4,[2,1]) .
3
Computation of α(G(w)) and β(G(w)). First consider the
V[w](cid:44){xij :i∈[q−1],j ∈[wi]}; digraph G(w). Observe that for i ∈ [q − 1],k ∈ [wi] we
C[w](cid:44)[q−1]; have in (x ,G(w)) = w − 1, out (x ,G(w)) = w − 1,
i ik i i ik
 (cid:32)[w ](cid:33) inj(xik,G(w))=wj andoutj(xik,G(w))=0forj (cid:54)=i.Con-
E[w](cid:44) ((xij,xij(cid:48)),i):i∈[q−1],(j,j)(cid:48) ∈ 2i ∪ siderGifors+1≤i≤q.Thenini(zi,Gi)=outi(yi,Gi)=1
  and all other indegrees and outdegrees are zero.
 (cid:32) (cid:33)  Let a = gcd(w ,w). Pick t = (cid:98)w/w (cid:99) so that 0 ≤ w −
 [q−1]  1 1
((xij,xi(cid:48)j(cid:48)),i):(i,i(cid:48))∈ 2 ,j ∈[wi],j(cid:48) ∈[wi(cid:48)] . tw1 <w1. Observe also that t≥s. Consider the vector
 
t
Let s be the largest integer such that w1 = w2 = ··· = ws. υ = w−atw1τ(x(t+1)1,G(w))+ wa1 (cid:88)τ(xi1,G(w)).
For each s+1≤i≤q−1, let G be anedge-colored digraph i=1
i
withtwoverticesy ,z andonedirectededgewithcolorifrom For j ∈[q−1], let in (υ) and out (υ) denote the coordinates
i i j j
y to z . Then G(w)={G(w)}∪{G :s+1≤i≤q−1}. in υ corresponding to the summation of the indegrees or
i i i
Example 1. Let w = [3,2]. The edge-colored digraph G(w) outdegrees with respect to color j. Then we have
is given below, where the solid lines denote directed edges wwj−w1 = w1(w−1), for j ∈[s],
with color 1, th(cid:111)(cid:111) e dot(cid:47)(cid:47)ted lin(cid:111)(cid:111)es den(cid:47)(cid:47)ote the directed edges with wwja−w1 < w1(wa−1), for s+1≤j ≤t,
color2,and“ ”(“ ”)denotesthetwodirected in (υ)= a a
edges with the same color with one in each direction. j wwwwtj+,1−aw+tw1 < wt+1(aw−1), foothrejrw=iste,+1,
a
x1(cid:85)(cid:85)1(cid:119)(cid:119)(cid:100)(cid:100)(cid:111)(cid:111) (cid:47)(cid:47)(cid:62)(cid:62)x1(cid:85)(cid:85)2 (cid:111)(cid:111) (cid:47)(cid:47)(cid:52)(cid:52)(cid:39)(cid:39)(cid:62)(cid:62)x13 out (υ)=w(w1−(watw−11))(,w−1) < w1(w−1), ffoorr jj ∈=[tt]+, 1,
j a a
0, otherwise.
Observe that the first 2s coordinates of υ are w (w −1)/a
1
and all other coordinates have values at most w (w−1)/a.
(cid:21)(cid:21) (cid:125)(cid:125) (cid:116)(cid:116) (cid:36)(cid:36) (cid:21)(cid:21) (cid:125)(cid:125) 1
x (cid:111)(cid:111) (cid:47)(cid:47)x Adding to υ a suitable nonnegative integral combinations of
21 22
τ(y ,G )’s and τ(z ,G )’s, we get w (w−1)/a(1,1,...,1).
(cid:47)(cid:47) i i i i 1
Then G2 is the digraph y2 z2 and the family of Hence, we conclude that α(G(w))=w1(w−1)/a.
digraphs is given by G(w)={G(w),G2}. Next, consider the edge vector µ(G) with G ∈ G(w). For
G(w), m (G(w)) = w (w−1) for i ∈ [q−1], while for G
Construction of an optimal CCC(n,2w−2,w) . Suppose i i i
q
with s+1 ≤ i ≤ q −1, m (G ) = 1 and m (G ) = 0 for
a superpure G(w)-decomposition of K(q−1) exists. For each i i j i
n j (cid:54)=i. Hence, β(G(w))=w (w−1).
F isomorphic to G(w), there is a unique partition V(F) = 1
(cid:83)q−1S so that the edges from x in F has color i if and only Applying Theorem 3, we obtain our first asymptotic result.
ifix=1∈Si . Then construct one codeword u with support V(F) Proposition 5. Fix w and let w =(cid:80)q−1w . There exists an
i i=1 i
such that u =i for x∈S . Hence, u has composition w. integer n such that
x i 0
Since we have a G(w)-decomposition of Kn(q−1), then (C1) n(n−1)
of Lemma 4 is satisfied. Furthermore, since the decompo- Aq(n,2w−2,w)= w (w−1)
1
sition is superpure, (C2) is also met. Hence, we obtain a
CCC(n,2w−2,w)q code of size n(n−1)/(w1(w−1)). The for all n≥n0 satisfying n(n−1)≡0 (mod w1(w−1)) and
code meets the upper bound in (1) and is therefore optimal. n−1≡0 (mod w1(w−1)/a), where a=gcd(w1,w).
B. When Distance d=2w−3 digraphs isomorphic to G∗(w). It is easy to see that M =
n(n−1)/(w (w −1)) if w (cid:54)= w and M = n(n−1)/w2,
We have the following analogous characterization of codes 1 1 1 2 1
otherwise. The code is optimal by the upper bound (2).
of constant-weight w with distance 2w−3.
We compute α(G∗(w)) and β(G∗(w)) in the full paper.
Lemma 6. ThefollowingaresufficientforacodeC ofweight
(cid:40)
w to have distance 2w−3. α(G∗(w))= w1(w1−1), if w1 >w2,
(C3) For i,j ∈ [q−1], the ordered pairs in the set {(x,y) : w1, otherwise.
u =i,u =j,x(cid:54)=y,u∈C} are distinct. (cid:40)
x y w (w −1), if w >w ,
(C4) For any u,v∈C, |supp(u)∩supp(v)|≤2. β(G∗(w))= 1 1 1 2
w2, otherwise.
1
Definition of G∗(w). For fixed w, define an edge-colored
Applying Theorem 3 we obtain the following proposition.
digraph G∗(w)=(V(w),C(w),E(w)), where
Proposition 7. Fix w and let w =(cid:80)q−1w . There exists an
i=1 i
V(w)(cid:44){xij :i∈[q−1],j∈[wi]}; integer n0 such that
C(w)(cid:44)[q−1]×[q−1]; (cid:40) n(n−1) , if w >w ,
(cid:40) (cid:41) A (n,2w−3,w)= w1(w1−1) 1 2
E(w)(cid:44) ((xij,xij(cid:48)),(i,i)):i∈[q−1],(j,j)(cid:48)∈(cid:16)[w2i](cid:17) ∪ q n(nw−121), otherwise,
(cid:40) (cid:41) for all n≥n satisfying
(cid:16)[q−1](cid:17) 0
((xij,xi(cid:48)j(cid:48)),(i,i(cid:48))):(i,i(cid:48))∈ 2 ,j∈[wi],j(cid:48)∈[wi(cid:48)] . (i) n−1≡0 (mod w1(w1−1)), if w1 >w2,
(ii) n−1≡0 (mod w2), otherwise.
1
For i,j ∈[q−1], let G∗ be a digraph with vertices y , z
ij i j IV. ASYMPTOTICALLYEXACTJOHNSON-TYPEBOUNDS
andonedirectededgewithcolor(i,j)fromy toz .Todefine
i j FORq-ARYCONSTANT-WEIGHTCODES
G∗(w), we have two cases depending on whether w =w :
1 2
In this section, we construct infinite families of optimal
(i) When w > w , let r be the largest integer such that
1 2 CWC(n,d,w) codes for d = 2w −2 or d = 2w −3, and
w = ··· = w = w − 1. Then set G∗(w) = q
2 r 1 establish the asymptotic exactness of (3) and (4).
{G∗(w)} ∪ {G∗ : (i,j) ∈ [q − 1] × [q − 1] \
ij Note that constant-composition codes are special instances
{(1,1),(1,2),(1,3),...,(1,r),(2,1),(3,1)...,(r,1)}.
ofconstant-weightcodes.Hence,wemakeuseofthedigraphs
(ii) When w = w , let r be the largest integer such that
1 2 definedinSectionIIItoformthedesiredfamiliesforconstant-
w = ··· = w . Then set G∗(w) = {G∗(w)} ∪
(cid:110)G1 ∗ :(i,j)∈[q−r1]×[q−1]\(cid:0)[r](cid:1)(cid:111). weight codes. Specifically, consider the set of compositions,
ij 2
 
Example 3. Let w =[3,2]. The edge-colored digraph G∗(w)  q(cid:88)−1 
(cid:47)(cid:47) (cid:47)(cid:47) W = [w1,w2,...,wq−1]:0≤wi≤w fori∈[q−1], wi=w .
is given(cid:47)(cid:47) below, where(cid:47)(cid:47) the lines “ ”, “ ”,  i=1 
“ ” and “ ” denote directed edges with color
Note here we do not require all values in the composition to
(1,1), (1,2), (2,1) and (2,2) respectively.
be positive and the composition to be monotone decreasing.
Furthermore,wecanextendtheconstructionsinSectionIIIto
(cid:119)(cid:119) (cid:39)(cid:39)
x1(cid:85)(cid:85)1 (cid:100)(cid:100)(cid:111)(cid:111) (cid:47)(cid:47)(cid:62)(cid:62)x1(cid:85)(cid:85)2 (cid:111)(cid:111) (cid:47)(cid:47)(cid:52)(cid:52)(cid:62)(cid:62)x13 define G(w) and G∗(w) for all w ∈W. Define the following
families of digraphs
(cid:91) (cid:91)
G(w)(cid:44) G(w) and G∗(w)(cid:44) G∗(w).
w∈W w∈W
(cid:21)(cid:21) (cid:125)(cid:125) (cid:116)(cid:116) (cid:36)(cid:36) (cid:21)(cid:21) (cid:125)(cid:125) Similar to constructions in Section III, we have that a
(cid:111)(cid:111) (cid:47)(cid:47)
x21 x22 superpure G(w)-decomposition of Kn(q−1) and a superpure
Then G∗22 is the digraph y2 (cid:47)(cid:47)z2 and the family of G∗(w)-decompositionofKn(q−1)2 yieldaCWC(n,2w−2,w)q
digraphs is given by G∗(w)={G∗(w),G∗ }. and a CWC(n,2w−3,w)q respectively.
22
ThenumberofdigraphsinasuperpureG(w)-decomposition
Construction of an optimal CCC(n,2w−3,w) . Suppose ofK(q−1)is(q−1)n(n−1)/(w(w−1))sincethetotalnumber
q n
a superpure G∗(w)-decomposition of K(q−1)2 exists. For F ofdirectededgesis(q−1)n(n−1).Inthefullpaperweused
n
isomorphic to G∗(w), there is a unique partition V(F) = themethodsofLamkenandWilsontoestablishthefollowing.
(cid:83)q−1S so that the edges from x to y in F has color (i,j)
i=1 i Proposition 8. Fix w. There exists an integer n0 such that
if x ∈ S and y ∈ S . Construct a codeword u with support
i j
(q−1)n(n−1)
V(SFin)cseucwhethahtauvxe=ai fsourpxerp∈uSrei. SGo∗,(wu)h-daseccoommppoossiittiioonn wof. Aq(n,2w−2,w)= w(w−1)
K(q−1)2, (C3) and (C4) of Lemma 6 are satisfied. Hence, we foralln≥n satisfying(q−1)n(n−1)≡0 (mod w(w−1))
n 0
obtain a CCC(n,2w−3,w) code. Let M be the number of and n−1≡0 (mod w−1).
q
On the other hand, G∗(w) corresponds to the family of di- The case for binary constant-weight codes, that is, q =2 and
graphsconstructedintheproofof[5,Theorem8.1].Therefore, d=2w−2, has been verified by Chee et al. [25].
we have the following proposition.
Proposition 9. Fix w. There exists an integer n such that APPENDIXA
0
ONG∗(w)ANDTHEPROOFOFPROPOSITION7
(q−1)2n(n−1)
A (n,2w−3,w)=
q w(w−1) Wecomputeα(G∗(w))andβ(G∗(w))definedinsectionIII.
foralln≥n satisfying(q−1)2n(n−1)≡0 (mod w(w−1))
0
A. When w >w .
and (q−1)(n−1)≡0 (mod w−1). 1 2
Recall r is the largest integer such that w = ··· = w =
V. ASYMPTOTICALLYEXACTJOHNSON-TYPEBOUNDS 2 r
w −1 and G∗(w)={G∗(w)}∪{G∗ :(i,j)∈[q−1]×[q−
FORMULTIPLYCONSTANT-WEIGHTCODES 1 ij
1]\{(1,1),(1,2),(1,3),...,(1,r),(2,1),(3,1)...,(r,1)}.
In this section, we construct an infinite family of optimal
Consider the digraph G∗(w). Observe that for i ∈
MCWC(m,n,2mw−2,w) and establish the asymptotic ex-
[q − 1] and k ∈ [w ], we have in (x ,G∗(w)) =
actness of (5). Fix m and w. Using digraphs from Section i (i,i) ik
out (x ,G∗(w)) = w − 1, in (x ,G∗(w)) =
III, we define the edge-colored digraph H∗(m,w) (cid:44) G∗(w), (i,i) ik i (j,i) ik
out (x ,G∗(w)) = w for j (cid:54)= i and and the other
with w = w = ··· = w = w (here q−1 = m). Define (i,j) ik j
1 2 m indegrees and outdegrees are zero. For the digraph G , we
H∗(m,w)={H∗(m,w)}∪{G∗ :i∈[m]}. ij
ii have out (y ,G ) = in (z ,G ) = 1 and the other
(i,j) ij ij (j,i) ij ij
Construction of an optimal MCWC(m,n,2mw − 2,w). indegrees and outdegrees are zero.
Suppose an H∗(m,w)-decomposition of K(m2) exists. For F Consider the vector
n
isomorphic to H∗(m,w), there is a unique partition V(F)=
(cid:83)m S so that the edges from x to y in F has color (i,j) if [(cid:88)q−1]
i=1 i υ =w τ(x ,G∗(w))+(w −1) τ(x ,G∗(w)).
1 11 1 i1
and only if x∈S and y ∈S . Then construct one codeword
i j
i=2
u such that u =1 for i∈[m] and x∈S .
(i,x) i
Since we have an H∗(m,w)-decomposition of Kn(m2), For (i,j) ∈ [q − 1] × [q − 1], let in(i,j)(υ) and out(i,j)(υ)
the distance of the code is 2mw − 2. Hence, we have an denote the coordinates in υ corresponding to the summation
MCWC(m,n,2mw−2,w) of size n(n−1)/w2 that meets of the indegrees and outdegrees with respect to color (i,j).
the upper bound given by (5). Similar computations yield Then we have
α(H∗(m,w)) = w and β(H∗(m,w)) = w2. Applying 
TPhroeporoesmitio3nw1e0.obFtaixinmtheanfdolwlo.wTinhge.re exists an integer n so ww11((ww11−−11)),, ffoorr ii==j1,=j (cid:54)=1,1,
0 in (υ)= w w for i(cid:54)=1,j =1,
that (i,j) 1 i
M(m,n,2mw−2,w)= n(nw−2 1) (ww(1w−1−)(1w),i−1) foothreirw=isje,,i(cid:54)=1,
i 1
for all n≥n0 satisfyVinIg.nC−ON1C≡LU0SI(OmNod w2). ww11(wwj1, −1), ffoorr ii==j1,=j (cid:54)=1,1,
out (υ)= w (w −1) for i(cid:54)=1,j =1,
We verify that Johnson-type bounds are asymptotically (i,j) 1 1
ecxoadcets.foTrhissevwearaslagcehnieevraeldizavtiiaonasnoifntberinesatriyngcoanpsptlaincat-twioenigohft (ww1(w−1−)(1w)i,−1) foothreirw=isje,,i(cid:54)=1,
j 1
superpure decompositions of edge-colored digraphs.
Observe that in Propositions 5, 7, 8, 9 and 10 Johnson-type Observe that the coordinates of υ corresponding to
boundsareshowntobeexactforsufficientlylargensatisfying indegrees and outdegrees with respect to colors in
certain congruence classes. We hypothesize that the Johnson- {(1,1),(1,2),(1,3),...,(1,r),(2,1),(3,1)...,(r,1)} have
typeboundsaretightuptoanadditiveconstantforsufficiently value w (w −1). All other coordinates have values at most
1 1
large n. Specifically, we make the following conjecture. w (w − 1). Adding to υ a suitable nonnegative integral
1
combinations of τ(y ,G )’s and τ(z ,G )’s, we obtain
ij ij ij ij
Conjecture. Fix q, w, w, m and let d ∈ {2w−2,2w−3}.
(w (w−1),w (w−1),...,w (w−1)). Hence, we conclude
1 1 1
DefineU (n,d,w),U (n,d,w),U(m,n,d,w)tobetheupper
q q that α(G)=w (w−1).
1
bounds given by (1) – (5). Then
Next, consider the edge vector µ(G) with G ∈ G∗(w).
Aq(n,d,w)=Uq(n,d,w)−O(1), For G∗(w), we have m(i,i)(G∗(w)) = wi(wi − 1) and
m (G∗(w)) = w w for j (cid:54)= i. On the other hand, for
A (n,d,w)=U (n,d,w)−O(1), (i,j) i j
q q G we have m (G ) = 1 and m (G ) = 0 for
ij (i,j) ij (i(cid:48),j(cid:48)) ij
M(m,n,2mw−2,w)=U(m,n,2mw−2,w)−O(1). (i(cid:48),j(cid:48))(cid:54)=(i,j). Hence, β(G∗(w))=w (w−1).
1
B. When w =w . since w(w−1) divides (q−1)n(n−1) and (6) holds. Then
1 2
(i) follows from Lemma 11.
Recall r is the largest integer such that w =···=w and
G∗(w)={G∗(w)}∪(cid:110)G :(i,j)∈[q−1]×1 [q−1]\(cid:0)[rr](cid:1)(cid:111). Proof of (ii): For i∈[q−1], let Xi and Yi be rationals
ij 2 such that integral conditions hold. In other words, for all G∈
Here, we consider the vector υ(cid:48) = (cid:80)qi=−11τ(xi1,G∗(w)). G(w) and u∈G, we have
Then the coordinates of υ(cid:48) corresponding to indegrees and
q−1
outdegrees with respect to colors in (cid:0)[r](cid:1) have value w . All (cid:88)
2 1 Xiini(u,G)+Yiouti(u,G)≡0.
other coordinates have values at most w . Adding to υ a
1 i=1
suitablenonnegativeintegralcombinationsofτ(y ,G )’sand
ij ij Fori∈[q−1],considerthedigraphG(w)withw =w and
τ(z ,G )’s,weconcludethatα(G∗(w))=w .Similartothe i
ij ij 1 w = 0 for j (cid:54)= i and consider any vertex in G(w). Hence,
case where w (cid:54)=w , we have β(G∗(w))=w2. j
1 2 1 we have (w−1)X +(w−1)Y ≡ 0. Since (w−1) divides
i i
APPENDIXB (n−1),
ONG(w)ANDTHEPROOFOFPROPOSITION8
(n−1)X +(n−1)Y ≡0.
i i
Unfortunately, it is not straightforward to determine
α(G(w)) and β(G(w)). Instead, we follow the methodology Therefore, the following relation is immediate.
ofLamkenandWilson(see[5,Theorem8.1])tocompletethe q−1
(cid:88)
proof of Proposition 8. Specifically, we show that provided (n−1) (X +Y )≡0,
i i
(q−1)n(n−1)≡0 (mod w(w−1))n−1≡0 (mod w−1),
i=1
we have:
Again (ii) follows from Lemma 11.
(i) the vector n(n − 1)(1,1,...,1) is an integral linear Proof of (iii): Summing up µ(G) for G ∈ G(w),
combination of vectors in {µ(G):G∈G(w)}; we obtain a constant vector of length q − 1 by symmetry.
(ii) the vector (n−1)(1,1,...,1) is an integral linear com- Therefore, admissibility of G(w) is immediate.
bination of vectors in {τ(u,G):u∈G,G∈G(w)};
REFERENCES
(iii) G(w) is admissible.
[1] R. M. Wilson, “An existence theory for pairwise balanced designs.
To establish (i) and (ii), we apply the following lemma.
I. Composition theorems and morphisms,” J. Combin. Theory Ser. A,
vol.13,pp.220–245,1972.
Lemma 11 (see Schrijver [26]). Let M be a rational m by n
[2] ——, “An existence theory for pairwise balanced designs. II. The
matrix and c be a rational vector of length m. Then Mx=c structureofPBD-closedsetsandtheexistenceconjectures,”J.Combin.
has an integral solution if and only if for all rational vectors TheorySer.A,vol.13,pp.246–273,1972.
y of length n, yTc is integral whenever yTM is integral. [3] ——,“Cyclotomyanddifferencefamiliesinelementaryabeliangroups,”
J.NumberTheory,vol.4,pp.17–47,1972.
[4] ——,“Anexistencetheoryforpairwisebalanceddesigns.III.Proofof
For convenience, we write a≡b if a−b is an integer.
theexistenceconjectures,”J.Combin.TheorySer.A,vol.18,pp.71–79,
Proof of (i): For i∈[q−1] let Xi be rationals such that 1975.
integralconditionshold.Inotherwords,forallG∈G(w),we [5] E. R. Lamken and R. M. Wilson, “Decompositions of edge-colored
completegraphs,”J.Combin.TheorySer.A,vol.89,no.2,pp.149–200,
have
2000.
q−1
(cid:88) [6] S. Hartmann, “Superpure digraph designs,” J. Combin. Des., vol. 10,
X m (G)≡0.
i i no.4,pp.239–255,2002.
i=1 [7] W.Chu,C.J.Colbourn,andP.Dukes,“Constructionsforpermutation
codesinpowerlinecommunications,”Des.CodesCryptogr.,vol.32,no.
For i ∈ [q −1], consider the digraph G(w) with wi = w 1-3,pp.51–64,2004.
and w =0 for j (cid:54)=i. Hence, we have [8] Y.M.Chee,H.M.Kiah,andP.Purkayastha,“Estimatesonthesizeof
j
symbolweightcodes,”IEEETrans.Inform.Theory,vol.59,no.1,pp.
w(w−1)X ≡0. (6) 301–314,January2013.
i [9] W.Chu,C.J.Colbourn,andP.Dukes,“Onconstantcompositioncodes,”
For (i,j) ∈ (cid:0)[q−21](cid:1), consider the digraph G(w) with wi = [10] DDi.sJc.reCteosAteplplol.aMndatGh..,Dvo.l.Fo1r5n4e,yn,o“.C6h,apnpn.el91co2d–i9n2g9:,t2h0e0r6o.adtochannel
w−1, w =1 and w =0 for k ∈/ {i,j}. capacity,”Proc.IEEE,vol.95,no.6,pp.1150–1177,2007.
j k
[11] Y. M. Chee, Z. Cherif, J.-L. Danger, S. Guilley, H. M. Kiah, J.-L.
(w−1)2X +(w−1)X ≡0. (7) Kim, P. Sole´, and X. Zhang, “Multiply constant weight codes and the
i j
reliabilityofloopphysicallyunclonablefunctions,”2013,preprint.
[12] M.Svanstro¨m,“Ternarycodeswithweightconstraints,”Ph.D.disserta-
Subtracting (7) from (6), we have (w−1)X ≡ (w−1)X
i j tion,Linko¨pingsUniversitet,Sweden,1999.
for all i,j. Since w−1 divides n−1, we have [13] M. Svanstro¨m, P. R. J. O¨sterga˚rd, and G. T. Bogdanova, “Bounds
andconstructionsforternaryconstant-compositioncodes,”IEEETrans.
Inform.Theory,vol.48,no.1,pp.101–111,2002.
(n−1)X ≡(n−1)X .
i j [14] F.-W. Fu, A. J. H. Vinck, and S.-Y. Shen, “On the constructions of
constant-weightcodes,”IEEETrans.Inform.Theory,vol.44,no.1,pp.
Finally, we compute 328–333,1998.
[15] Y.M.Chee,G.Ge,andA.C.H.Ling,“Groupdivisiblecodesandtheir
q−1
(cid:88) application in the construction of optimal constant-composition codes
n(n−1) Xi ≡(q−1)n(n−1)X1 ≡0, ofweightthree,”IEEETrans.Inform.Theory,vol.54,no.8,pp.3552–
3564,2008.
i=1
[16] Y. M. Chee, S. H. Dau, A. C. H. Ling, and S. Ling, “Linear size
optimal q-ary constant-weight codes and constant-composition codes,”
IEEETrans.Inform.Theory,vol.56,no.1,pp.140–151,2010.
[17] M. Svanstro¨m, “Constructions of ternary constant-composition codes
with weight three,” IEEE Trans. Inform. Theory, vol. 46, no. 7, pp.
2644–2647,2000.
[18] F. Gao and G. Ge, “Optimal ternary constant-composition codes of
weight four and distance five,” IEEE Trans. Inform. Theory, vol. 57,
no.6,pp.3742–3757,2011.
[19] Y. M. Chee and S. Ling, “Constructions for q-ary constant-weight
codes,”IEEETrans.Inform.Theory,vol.53,no.1,pp.135–146,2007.
[20] Y.M.Chee,S.H.Dau,A.C.H.Ling,andS.Ling,“Thesizesofoptimal
q-ary codes of weight three and distance four: a complete solution,”
IEEETrans.Inform.Theory,vol.54,no.3,pp.1291–1295,2008.
[21] Y. M. Chee, G. Ge, H. Zhang, and X. Zhang, “Hanani triple packings
andoptimalq-arycodesofconstantweightthree,”2013,preprint.
[22] H.ZhangandG.Ge,“Optimalternaryconstant-weightcodesofweight
fouranddistancesix,”IEEETrans.Inform.Theory,vol.56,no.5,pp.
2188–2203,2010.
[23] H.Zhang,X.Zhang,andG.Ge,“Optimalternaryconstant-weightcodes
with weight 4 and distance 5,” IEEE Trans. Inform. Theory, vol. 58,
no.5,pp.2706–2718,2012.
[24] H. Zhang and G. Ge, “Optimal quaternary constant-weight codes with
weight four and distance five,” IEEE Trans. Inform. Theory, vol. 59,
no.3,pp.1617–1629,2013.
[25] Y.M.Chee,C.J.Colbourn,A.C.H.Ling,andR.M.Wilson,“Covering
and packing for pairs,” J. Comb. Theory Ser. A, vol. 120, no. 7, pp.
1440–1449,2013.
[26] A. Schrijver, Theory of linear and integer programming. Chichester:
JohnWiley&Sons,1986.