Constructing c -ary perfect factors

TLDR: This paper conjecture that the same is true for arbitrary values ofc, and exhibit a number of constructions that construct a family of related combinatorial objects, which are called Perfect Multi-factors.

Abstract: Ac-ary Perfect Factor is a set of uniformly long cycles whose elements are drawn from a set of sizec, in which every possiblev-tuple of elements occurs exactly once. In the binary case, i.e. wherec=2, these perfect factors have previously been studied by Etzion [2], who showed that the obvious necessary conditions for their existence are in fact sufficient. This result has recently been extended by Paterson [4], who has shown that the necessary existence conditions are sufficient wheneverc is a prime power. In this paper we conjecture that the same is true for arbitrary values ofc, and exhibit a number of constructions. We also construct a family of related combinatorial objects, which we callPerfect Multi-factors.

Full text

Constructing c-ary Perfect Factors
Chris J. Mitchell
Computer Science Department,
Royal Holloway, University of London,
Egham Hill,
Egham,
Surrey TW20 0EX,
England.
Tel.: +44–784–443423
Fax: +44–784–443420
Email: cjm@dcs.rhbnc.ac.uk
20th March 1993
Abstract
A c-ary Perfect Factor is a set of uniformly long cycles whose ele-
ments are drawn from a set of size c, in which every possible v-tuple
of elements occurs exactly once. In the binary case, i.e. where c = 2,
these perfect factors have previously been studied by Etzion, [2], who
showed that the obvious necessary conditions for their existence are in
fact sufficient. This result has recently been extended by Paterson, [4],
who has shown that the necessary existence conditions are sufficient
whenever c is a prime power. In this paper we conjecture that the same
is true for arbitrary values of c, and exhibit a number of constructions.
We also construct a family of related combinatorial objects, which we
call Perfect Multi-factors.
1

Index Terms: de Bruijn graph, de Bruijn sequence, window se-
quence.
1 Introduction
Perfect factors were introduced, in the binary case, by Etzion, [2], who used
them to construct a certain class of (binary) Perfect Maps. In doing so
Etzion succeeded in showing that all the possible binary Perfect Factors
exist. In this paper we are concerned with Perfect Factors over arbitrary
finite alphabets. The motive for constructing these objects is two-fold.
Firstly, they can b e used in an obvious generalisation of Etzion’s construction
to construct non-binary Perfect Maps; for further details see [4]. Perfect
Maps, both binary and non-binary, have possible application in the field of
automatic position sensing, see, for example, [1].
Secondly, they are of interest in their own right as natural generalisations
of the classical de Bruijn sequences, about which much has been written.
As described in [4], they also have applications in other areas, including the
construction of de Bruijn sequences with minimal linear complexity.
1.1 Preliminary remarks and notation
We are concerned here with c-ary periodic sequences, where by the term c-
ary we mean sequences whose elements are drawn from the set {0, 1, . . . , c −
1}. We refer throughout to c-ary cycles of period n, by which we mean
cyclic sequences (s
0
, s
1
, . . . , s
n−1
) where s
i
∈ {0, 1, . . . , c − 1} for every i,
(0 ≤ i < n).
If t = (t
0
, t
1
, . . . , t
v−1
) is a c-ary v-tuple (i.e. t
i
∈ {0, 1, . . . , c−1} for every i,
(0 ≤ i < v)), and s = (s
0
, s
1
, . . . , s
n−1
) is a c-ary cycle of period n (n ≥ v),
2

then we say that t occurs in s at position j if and only if
t
i
= s
i+j
for every i, (0 ≤ i < v), where i + j is computed modulo n.
If s and s
′
are two v-tuples, then we write s + s
′
for the v-tuple obtained by
element-wise adding together the two tuples. Similarly, if k is any integer,
we write ks for the tuple obtained by element-wise multiplying the tuple
s by k. Again, if we write t = s mod k, then t is the tuple obtained by
reducing every element in s modulo k. An exactly analogous interpretation
should be used for arithmetic operations on cycles.
Given a cycle s = (s
i
), (0 ≤ i < n), and any integer k, we define T
k
(s) to
be the cyclic shift of s by k places. I.e. if we write s
′
= (s
′
i
) = T
k
(s) then
s
′
i+k
= s
i
, (0 ≤ i < n)
where i + k is calculated modulo n.
Suppose u = (u
0
, u
1
, . . . , u
n−1
) and u
′
= (u
′
0
, u
′
1
, . . . , u
′
n
′
−1
) are c-ary cycles
of periods n and n
′
respectively. Then define the concatenation of u and u
′
,
written
u||u
′
to be a c-ary cycle of period n + n
′
s = (s
0
, s
1
, . . . , s
n+n
′
−1
) = u||u
′
,
where
s
i
=





u
i
if 0 ≤ i < n
u
′
i−n
if n ≤ i < n + n
′
We also need some notation linking sets of c-ary cycles with matrices.
Suppose that A = {a
0
, a
1
, . . . , a
c
v
−1
} is a set of c
v
c-ary cycles of pe-
riod n. Then let X
A
be the c
v
× n matrix with row i equal to a
i
, (0 ≤
3

i ≤ c
v
− 1). Conversely, suppose that X is a c
v
× n matrix. Then let
A
X
= {a
X
0
, a
X
1
, . . . , a
X
c
v
−1
} be the set of c-ary cycles (of length n) defined so
that a
X
i
is equal to row i of X (0 ≤ i ≤ c
v
− 1).
We use the following matrix notation. Suppose X and Y are matrices of
dimensions s × t and s × u resp ectively. Then Z = (X |Y) denotes the
s × t + u matrix whose first t columns are the columns of X and whose last
u columns are the columns of Y. For any matrix X , the transpose of X is
denoted by X
T
.
Finally note that, throughout this paper, the notation (m, n) represents the
Greatest Common Divisor of m and n (given that m, n are a pair of positive
integers).
1.2 Fundamentals
We can now define the combinatorial objects which are the main focus of
this paper.
Definition 1.1 Suppose n, c and v are positive integers (where we also
assume that c ≥ 2). An (n, c, v)–Perfect Factor, or simply a (n, c, v)–PF, is
then a set of c
v
/n c-ary cycles of period n with the property that every c-ary
v-tuple occurs in one of these cycles.
Note that, because we insist that a Perfect Factor contains exactly c
v
/n
cycles, and because there are clearly c
v
different c-ary v-tuples, each v-tuple
will actually occur exactly once somewhere in the set of cycles. Also observe
that a (c
v
, c, v)–PF is simply a c-ary span v de Bruijn sequence.
Example 1.2 The following three cycles form a (3, 3, 2)–PF.

0 0 1

,

1 1 2

,

2 2 0

.
4

The following necessary conditions for the existence of a Perfect Factor are
trivial to establish.
Lemma 1.3 Suppose A is a (n, c, v)–PF. Then
1. n|c
v
, and
2. v < n ≤ c
v
.
We now state our main conjecture regarding the existence of Perfect Factors.
Conjecture 1.4 The necessary conditions specified in Lemma 1.3 are suf-
ficient for the existence of a Perfect Factor.
Etzion, [2], showed that Conjecture 1.4 is true in the binary case, i.e. c = 2.
Paterson, [4], has recently shown that Conjecture 1.4 is true whenever c = p
α
for p any prime and α a positive integer. In this paper, as an effort towards
establishing this conjecture, we demonstrate some constructions for c-ary
Perfect Factors for general c.
2 Perfect Multi-factors
Before giving our first method of construction for Perfect Factors, we define
a related set of combinatorial objects which will be of some use in their
construction.
Definition 2.1 Suppose m, n, c and v are positive integers satisfying m|c
v
and c ≥ 2. An (m, n, c, v)–Perfect Multi-factor, or simply a (m, n, c, v)–
PMF, is a set of c
v
/m c-ary cycles of period mn with the property that for
every c-ary v-tuple t and for every integer j in the range 0 ≤ j < n, t occurs
at a position p ≡ j (mod n) in one of these cycles.
5

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Constructing c-ary perfect factors" ?

In the binary case, i. e. where c = 2, these perfect factors have previously been studied by Etzion, [ 2 ], who showed that the obvious necessary conditions for their existence are in fact sufficient. In this paper the authors conjecture that the same is true for arbitrary values of c, and exhibit a number of constructions. 

Q2. What is the property of v t in the v v ?

If the v × t matrix X has the property that every v × v sub-matrix of (Iv|X |Iv) is invertible in the ring of v× v matrices modulo c, the authors say that X has Property X.Lemma 3.12 Suppose c ≥ 2, v and t are positive integers. 

Q3. What is the c-ary v dmatrix y?

If d > 0 then, by the inductive hypothesis, there exists a c-ary v × dmatrix Y such that every v × v sub-matrix of (Iv|Y|Iv) is invertible. 

Q4. What is the property of the t t matrix?

If v = t = 1 then D = (1) trivially has Property X. Now suppose a matrix with Property X exists for every v, t satisfying max{v, t} < L for some positive integer L > 1. 

Q5. What is the c-ary cycle of n?

I.e. if the authors write s′ = (s′i) = Tk(s) thens′i+k = si, (0 ≤ i < n)where i+ k is calculated modulo n.Suppose u = (u0, u1, . . . , un−1) and u ′ = (u′0, u ′ 1, . . . , u ′ n′−1) are c-ary cycles of periods n and n′ respectively. 

Q6. What are the conditions for the existence of a Perfect Factor?

Note that, because the authors insist that a Perfect Factor contains exactly cv/n cycles, and because there are clearly cv different c-ary v-tuples, each v-tuple will actually occur exactly once somewhere in the set of cycles. 

Q7. what is the condition for the existence of a perfect multi-factor?

Note that, because the authors insist that a Perfect Multi-factor with m = 1 contains exactly cv cycles, and because there are clearly cv different c-ary v-tuples, each v-tuple will actually occur exactly n times in the set of cycles, once in each of the possible positions. 

Q8. What are the c-ary cycles of period n?

The authors refer throughout to c-ary cycles of period n, by which the authors mean cyclic sequences (s0, s1, . . . , sn−1) where si ∈ {0, 1, . . . , c − 1} for every i, (0 ≤ i < n). 

Q9. what is the c-ary vector of length v+ t?

for every j and s, (0 ≤ j ≤ v − 1, 0 ≤ s ≤ v + t − 1), there exist integers e (s) ij , (0 ≤ i ≤ v − 1), such thatxj = v−1∑ i=0 e (s) ij xs+i mod cwhere the subscript s+ i is computed modulo v+ t, if and only if every v× v sub-matrix of (Iv|D|Iv) is invertible over the ring of v × v matrices modulo c (where Iv is the v × v identity matrix). 

Q10. What is the t t sub-matrix of X T?

Since t < s ≤ v, this consists of columns s − t, s − t + 1, . . . , s − 1 of X T , and hence this t× t sub-matrix of X T is invertible. 

Q11. what is the c-ary matrix of dimensions v t?

Further suppose that X is a c-ary matrix of dimensions cv × v having column vectorsxT0 ,x T 1 , . . . ,x T v−1whose first v rows are equal to Iv, and that Y is a c-ary matrix of dimensions cv × t having column vectorsxTv ,x T v+1, . . . ,x T v+t−1whereY = XD mod c (1)and D = (dij) (0 ≤ i ≤ v−1, 0 ≤ j ≤ t−1), is a c-ary matrix of dimensions v × t. 

Q12. what is the 'trivial' partition of the cycles of A?

Let A0 = {A0,0, A0,1, . . . , A0,cv−1} be the ‘trivial’ partition of the cycles of A defined byA0,i = {ai}for every i, (0 ≤ i < cv). 

Q13. What is the concatenation of u and u′?

Then define the concatenation of u and u′, writtenu||u′to be a c-ary cycle of period n+ n′s = (s0, s1, . . . , sn+n′−1) = u||u′,wheresi = ui if 0 ≤ i < nu′i−n if n ≤ i < n+ n′ 

Q14. what is the mn cycle of a cycle?

given m = 2, the concatenated cycles w0, w1, w2 are as follows:w0 = ( 0 0 1 0 0 1 ) , w1 = ( 1 1 2 1 1 2 ) , w2 = ( 2 2 0 2 2 0 ) . 

Q15. what is the v-partition of the cycles of a?

Hence if the authors apply Algorithm 4.11 v times to A0 to obtain the partitionAv = {Av,0, Av,1, . . . , Av,qv−1},then, by Lemma 4.12, Av is an (v, rv, v)–Partition of the cycles of A.