1988
IEEE
TRANSACTIONS
ON. INFORMATION THEORY,
VOL.
39,
NO.
6,
NOVEMBER
1993
Minimal-Change Order and Separability in Linear
Codes
A.
J.
van Zanten
Abstract-A
linear code
E?
is said to he in minimal-change order if
each codeword differs from
its
predecessor by a word of minimum
weight.
A
rule is presented to construct such an order in case that
i?
has
a basis of codewords with minimum weight. Some consequences concem-
ing the ranking and separability in
5F
are mentioned.
Index
Terms-Minimal-change order, Gray codes, ranking problem,
separability.
I.
PRELIMINARIES
It is well known that the set of all binary words of length
n
can be ordered in a list such that each word differs from its
predecessor by precisely 1 bit. Such a list is called a Gray code.
For any value of
n,
there are many such lists possible. The best
known example is the so-called
binary rejlected
or
normal Gray
code
(cf.
[3,
pp. 172-1771]. We denote this code by the matrix
r
1
(1)
If
we write
g,
:=
g,,-lg,,-2
...
g,",
and if
b,
=
b,,-lb,,_2
...
b,,,
is the binary representation of the index
i,
then the following
rules hold for
0
I
i
<
2" and
0
I
j
<
n:
and
n-
1
b,,
=
g,,
(mod21 (3)
I=]
with
b,,,
:=
0
(cf., e.g., [3]). Rules (2) and (3) solve the
ranking
problem
of
G(n).
A
related question in this context is the
separabilityproblem.
If
two
codewords
g,
and
gl
have Hamming distance
m,
one
can
ask how they are located with respect to each other in the
ordered list
G(n)
or, more specifically, one can ask for bounds
for their
Gray distance Ii
-
jl.
Actually, one has
[2"/3]
I
Ii
-jl
I
12"
-
2"'/31.
(4)
Both bounds are sharp. The lower bound was derived by Yuen
[6] and the upper bound by Cavior in [l].
Similar results have been obtained for the
constant weight
Gray code
G(n,
k),
consisting of words of length
n
and weight
k
(cf. [7]).
In
this code, each word differs from its predecessor by
precisely 2 bits.
Both
G(n)
and
G(n,
k)
are examples of lists ordered by a
minimal-change principle.
There are many more combinatorial
objects (permutations, compositions, graphs, etc.) which can be
ordered according to a minimal-change principle. For a review,
we refer
to
[5].
Manuscript received July 30, 1992.
The author is with the Faculty
of
Technical Mathematics and Infor-
matics, Delft University
of
Technology,
2600
GA Delft, The Nether-
lands.
IEEE
Log
Number 9212919.
In
Section
11,
we show that there is also a wide class
of
linear
error-correcting codes,
which can be ordered in this way.
11.
LINEAR CODES
IN
MINIMAL-CHANGE ORDER
Let
F?
be an arbitrary linear binary code, and let this code be
ordered. We shall say that
E'
is in a minimal-change order, or is
ordered according to a minimal-change principle,
if
each code-
word differs from its predecessor by a word of minimum weight.
A
necessary and sufficient criterion for the existence of such an
order in a given code can easily be given. It is the special case
with
w
=
d
of the following theorem.
Theorem:
A
linear
[n,
k,
d]-code
55'
can
be
ordered such that
each codeword differs from its predecessor
by
precisely w bits
if
and only if
E'
has a basis of codewords by weight w.
Proof:
It is obvious that the condition is necessary, since the
existence
of
the described order implies that
E'
can be gener-
ated by codewords of weight
w.
The condition is also sufficient. To show this, we assume that
A
:=
(a",
a,;..,
ak- is a basis, such that each
a,
has weight
w,
for
0
I
j
I
k
-
1.
Let
g,
be
the ith codeword
of
the normal
Gray code
G(k)
(cf. Section
I).
We define
k-
1
,=O
c,
=
glial,
0
I
i
<
2k.
(5)
If
i
runs through its value set, we obtain all linear combina-
tions of the words of
A.
Furthermore, since
g,
and
gL+l
differ
by
1
bit, it follows that
c,
and
c,+~
differ by one word of the
basis
A,
which has weight
w,
for all relevant values of
i.
This
From
now
on,
we assume that
A
is a basis of words
of
minimum weight
d,
and furthermore, that the code
55'
is in
minimal-change order according to the construction in the above
proof. The ranking problem for
F?
can easily be solved. To
determine
c,
for
a
given value of
I,
one first converts
i
to its
Gray representation, using (2), and next applies
(5).
As an
alternative, one can equally well use the generator matrix
completes the proof.
0
-
-
G(55')
=
and apply the relation
ci
=
b,G(t?),
(6)
(7)
which directly converts the binary representation of
i
to the ith
codeword of
g.
Finally, we present bounds for the Gray distance in
55'.
These
follow from the inequalities (4). If
c,
and
c,
are codewords
of
the ordered code
g
with Hamming distance
m,
then
(8)
where
m'
:=
[m/dl.
Whether these bounds are sharp depends
on
the code
E'
and
on
the chosen basis
A.
Remark:
There are many codes that have a basis of words of
minimum weight, e.g., Hamming codes and Reed-Muller codes.
Furthermore, all codes that meet the Griesmer bound have such
a basis (cf. [2], [4]).
/2"'/3]
5
(i
-
jl
I
12"
-
2"'/31
REFERENCES
[l]
[2]
S.
R.
Cavior,
"An
upper bound associated with errors in Gray
code,"
IEEE
Trans.
Inform.
Theory,
vol. IT-21,
p.
596,
1975.
S.
M. Dodunekov and
N.
L.
Manev,
"An
improvement
of
the
Griesmer bound for small minimum distances,"
Discrete
Appl.
Math.,
vol.
12,
pp. 103-114, 1983.
0018-9448/93$03.00
0
1993
IEEE
E.
M.
Reingold,
J.
Nievergelt, and
N.
Deo,
Combinatorial
Algo-
rithms:
Theory and Practice.
Englewood Cliffs,
NJ:
Prentice-Hall,
1977.
H.
van Tilborg,
“On
the uniqueness resp. nonexistence
of
certain
codes meeting the Griesmer bound,”
Discrete Math.,
vol.
44,
pp.
H.
S.
Wilf, “Combinatorial algorithms:
An
update.” in
CBMS-NSF
Regional Conf. Ser.
Appl.
Math.,
Soc. Indust. Appl. Math., Philadel-
phia, PA, 1989.
C.
K.
Yuen,
“The separability of Gray code.”
IEEE Trans. Znform.
Theory,
vol. IT-20, p. 668, 1974.
A.
J.
van Zanten, “Index system and separability
of
constant weight
Gray codes,”
IEEE Trons. Inform. Theory,
vol. 37, pp. 1229-1233,
1991.
16-35, 1980.
The Number
of
Nonlinear Shift Registers That
Produce
All
Vectors
of
Weight
I
t
Harold Fredricksen
,
Member,
IEEE
Abstract-It has been shown that it is possible to generate a cycle on a
nonlinear shift register to contain all vectors of length
n
and Hamming
weight
5
t. We show how
to
count the number of different ways this can
be done on a truth table of minimum density.
Index
Terms-BEST
theorem, binary sequence, de Bruijn graph, cycle
joining, shift register, spanning tree, truth table.
I.
INTRODUCTION
In
[l],
it is shown that a nonlinear shift register can be
designed to generate all of the vectors of length
n
having no
more than a Hamming weight of
t
ones. Applications for such
functions are discussed there. Examples for all
n
I
7
and
1
I
t
I
n
are given. The authors also give an example
of
a
shift
register feedback function to produce one such sequence, with
each binary vector appearing exactly once for each appropriate
n
and
t.
Similar approaches were used [2]-[4] to connect all
vectors of length
n
into a single de Bruijn cycle. In this note, we
show in how many ways cycles of a given length and weight may
be joined for a certain subclass of shift register feedback func-
tions. Our class is that for which each feedback truth table
contains the minimum possible density of ones.
In order to ensure that any feedback function
f(xo,x1;..,
x,,-
produces only cycles, it suffices that
f
be of the form
f
=
xg
+
g(x,;..,
x,-
,)
[5].
The feedback function
f
=
xu
is of
the branchless type with
g(xl;..,
xu-
0
and
is
called the
pure cycling register. The factor created consists of cycles de-
fined as the cyclotomic cosets of vectors formed as equivalence
classes under the cyclic rotation of the bits of the vectors of
GF(2)“. The density of the truth table of this pure cycling
register
is
zero. The cycles formed are enumerated by
Z(n),
Z(n)
=
(l/n)C,,n,&(d)2”/d where the summation is over all
positive integer divisors
d
of
n
and
&
is Euler’s totient function.
The cycles are all cycles of vectors of length
d
for each divisor
d
of
n.
These cycles form the fundamental building blocks for the
theory developed in
[l].
Here, we note simply that two of these
Manuscript received
December
10.
1992,
revised
Februaty 16,
1993.
The author was
on
leave at the Institute
for
Defense Analyses, Center
for
Communications Research, San Diego,
CA
92121. He was with the
Department of Mathematics, Naval Postgraduate School, Monterey, CA
93943.
IEEE Log
Number 9213028.
pure cycles may be joined together
if
there is an adjacency
between them.
We refer the reader to
[l],
[3]-[5]
for any additional required
shift register theory. The cycle decompositions of the underlying
de Bruijn graph, cycle adjacencies, and joins are well covered
there and need not be duplicated here.
11.
SEQUENCES
OF
VECTORS
OF
WEIGHT
I
t
The sequences of
[l]
are formed by joining all
of
the cyclo-
tomic cycles
of
vectors of length
n
and having
5
t
ones. If there
are
C
such cycles, they may be joined together in several ways,
but this surgery will always require that at least
C
-
1
positions
of the truth table of
g
be changed from zero to one. If we
restrict the truth tables to have no more than a density of
C
-
1
ones, then we can count the number of ways to generate such a
sequence. We employ a result of de Bruijn, Ehrenfest, Smith,
and Tutte described in
[6]-[8],
and used in
[4]
to generate de
Bruijn cycles by joining all of the cycles of the pure cycling
register together.
Theorem
I
(BEST):
The number of spanning subtrees of a
labeled connected graph is evaluated by computing the determi-
nant of the cofactor of a root in the associated adjacency matrix
of the graph.
To use the theorem to find the number of sequences we seek,
we call each cyclotomic coset a node in a graph, and we label the
edge between cosets
C,
and
C,
with the number of vectors on
C,
having a conjugate on
C,.
The problem of adjacencies in the
pure cycling register has been studied previously
[9],
[
101.
Then,
the spanning subtrees give the smallest number of changes to
g
that can be made to join all of the cycles into a single cycle. It
may be possible to join cycles in another way by splitting the
cycle created and then rejoining the pieces in a different way.
We do not include these cycles in our count.
Two examples of cycles formed in this way are given in
[
11.
Sec
also
[4],
where all of the pure cycles are joined to form a de
Bruijn cycle. For the case at hand, only the subset of the cycles
of weight
I
t
are used. Then, the subgraph
H(n,w)
is con-
structed.
H(n,w)
is
obtained from the graph
G(n)
of all pure
cycles and their edges under adjacency by removing all vertices
of
weight bigger than
w.
Thus, the vertices of the graph
H(n,
t)
are the cycles of the pure cycling register with weight less than
t
+
1.
There is an edge in
H(n,
t)
between nodes
x
and
y
if
there is an adjacency between the cycles represented.
The BEST theorem states that the number of spanning trees
of these graphs is found by computing the determinant of the
cofactor of
a
root of the graph. In all such graphs, the node
0
will be a root,
so
we need only evaluate the determinant of the
cofactor of the
(0,O)
entry of the matrix.
The matrix size grows with
n
and
t,
but the determinants are
easily evaluated with Maple. In Table I, the factored forms of
the determinants are given.
No
obvious patterns emerge for the
cosets of weight
t
=
3
and higher. For any
n,
when
t
=
1,
there
is obviously only one way to add the coset
(00
...
01)
to the coset
(0).
For
t
=
2,
there is a twofold choice of how to add each of
the cosets of weight
2
when
n
is odd. When
n
is even, there
is
also a coset of size
n/2
which can be added to the cycle in a
unique way. Hence, the number
N(n,
2)
=
2[’f-2/21.
111. CONCLUSIONS
We have shown how
a
theorem on spanning subtrees
on
a
graph can be used to evaluate the number of sequences that
IEEE
TRANSACTIONS
ON
INFORMATION THEORY,
VOL.
39,
NO.
6.
NOVEMBER 1993
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1993
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