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2005
Multinomial Representation of Majority Logic Coding Multinomial Representation of Majority Logic Coding
John B. Moore
Australian National University
Keng T. Tan
Edith Cowan University
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10.1109/ISIT.2005.1523368
This is an Author's Accepted Manuscript of: Moore, J.B., & Tan, K.T. (2005). Multinomial Representation of Majority
Logic Coding. Proceedings of IEEE International Symposium on Information Theory. ISIT 2005 (pp. 421-424).
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Multinomial Representation of Majority Logic
Coding
John B. Moore
Research School of Info. Sci. & Eng.
Australian National University
Canberra ACT 0200, Australia
Email: john.moore@anu.edu.au
Keng T. Tan
School of Computer and Information Science
Edith Cowan University
Mount Lawley WA 6050, Australia
Email: a.tan@ecu.edu.au
Abstract— Multinomial representations are derived for major-
ity logic operations on bipolar binary data. The coefficients are
given simply in terms of the readily computed lower Cholesky
factor of Pascal Matrices of order n for codes of block length n.
I. INTRODUCTION
Majority voting on binary data is the basis of certain non-
linear block coding schemes in communication systems [1],
especially in the case where, extremely low power radio wave
communications is desired [2]. The majority logic operation
is used in both the coding and decoding operations. Because
of the nonlinearity of the operation, there is difficulty in
predicting system performance, or seeing how to improve
system performance. Our view is that a crucial tool in this task
is a multinomial representation of the majority logic operation.
A multinomial expansion for majority logic has been par-
tially studied in [3], [4], and the results applied in various
communication contexts. General formulas for the first and
last coefficients in the expansion are stated, and for bipolar
binary vectors of length n, it is claimed that the even numbered
coefficients are zero for n even, but we know of no sources
which give other coefficients.
Here, we give a complete theory for the multinomial repre-
sentations of majority logic operations on bipolar binary data.
The majority logic operation can be a classic sign function of
the sum of the binary data, as studied in the earlier literature
known to us. Perhaps more usefully, we also give a theory for
what we term here sign
±
functions. These are sign operations
where an output of 0 is replaced by ±1. The approach extends
to other nonlinear functions of the sum of binary data, such
as to sigmoidal functions used in artificial neural networks. It
also extends to arbitrary nonlinear functions of bipolar binary
data vectors that are invariant of the order of the data within
the vector.
The coefficients of the multinomial expansion are linear
in what we call a generalized Pascal matrix, which can be
factored in terms of the lower triangular Cholesky factor,
denoted here P
n
,ofaPascal matrix of order n.The‘new’
results are generalizations of the classical results. It would not
be surprising if at least some of the results were known by
Pascal, but the motivation for deriving them, or highlighting
them, is coming from applications of nonlinear coding for next
generation wireless communications.
In Section II, the new results on multinomial expansions
of majority logic functions are derived. In Section III, these
results are applied to majority logic based nonlinear block
coding. Conclusions are drawn in Section VI.
II. T
HE PASCAL MATRIX AND A MULTINOMIAL EXPANSION
In this section, we introduce background material on clas-
sical results in order to set up notation for the main results of
the following sections.
Our results concern nonlinear operations on a data n-vector
a =[a
1
,a
2
,...,a
n
]
with a
i
∈{+1, −1}. Now any nonlinear
function of a belongs to a finite discrete set of no more than
2
n
elements. Indeed, such functions are linear in an indicator
2
n
-vector ∈{e
1
,e
2
,...,e
2n
},wheree
i
is a zero 2
n
-vector
save that the i
th
element is unity.
Our new results concern nonlinear operations that are invari-
ant of any ordering in the data, such as functions of
n
i=1
a
i
,
n
i=1
a
i
,orof
n
i=1
(1+a
i
). In this case, the functions belong
to a discrete set of at most n +1 elements.
Our focus is on (nonlinear) majority logic functions involv-
ing sign operations on sums of partial products a
i
,whichmap
one-to-one to the data vector. The resulting representations are
termed multinomial representations.
A. Multinomial representation for majority logic
1) Nonlinear functions and majority logic: Consider the
sign function definition.
sign(x):=
1ifx>0
0ifx =0
−1ifx<0
. (1)
Let us also introduce derivative definitions, denoted sign
+
and
sign
−
as
sign
±
(x):=
1ifx>0
±1ifx =0
−1ifx<0
. (2)
Consider now a set of n bipolar binary digits
{a
1
,a
2
,...,a
n
}, that is where a
i
∈{+1, −1}.The
majority logic operation on this n-block of data is simply
sign
∗
(
n
1
a
i
), where we have used sign
∗
to denote either
sign, sign
+
or sign
−
. The latter two options can be used
if the output of the logic operation is constrained to be also
bipolar binary.
2) Multinomial representation: Early literature [3][4],
presents a multinomial representation for the majority logic
sign operation, which we here also mildly generalize as
sign
∗
n
i=1
a
i
= ρ
0
+ ρ
1
n
i=1
a
i
+ ρ
2
all i>j
a
i
a
j
+ρ
3
all i>j>k
a
i
a
j
a
k
+ ···+ ρ
n
n
i=1
a
i
. (3)
for suitable selections of coefficients ρ := [ρ
0
,ρ
1
,...,ρ
n
]
,
which will depend on which of the sign operations is used.
The selection of the coefficients and their properties is the
study of this paper.
Of course, the expansion of the nonlinear function
n
i=1
(1+
a
i
) has such an expansion as the right hand side of (3) with
coefficients ρ =[1 1...1]. The mapping from the set {a
i
}
to the set of the sums of products in (3) via the coefficients
of the ρ
i
, is known to be one to one.
The earlier work has given specific formulas for the coef-
ficients ρ
0
,ρ
1
,ρ
n
of (3) in terms of permutation operations
n
C
i
=
n!
i!(n−i)!
, at least for the case of the classic sign
function. It is also noted in the early work that in this case
ρ
i
=0for n, i even, but other coefficients have not been
studied to our knowledge.
In our applications of such expansions, it is important to
have readily calculated coefficients for all the coefficients ρ
i
,
and to see relationships between them in order to understand
experimentally observed relationships in majority logic coding
for communication systems.
In order to proceed, we first review relevant results of the
Pascal matrix.
B. The lower Cholesky factor of the Pascal matrix
The well known (second) Pascal matrix, is a lower
Cholesky factor of the original (first) Pascal matrix. We will
refer to this (second) Pascal matrix simply as the P ascal
matrix, and use the notation P
n
=(p
i,j
n
) for such an n × n
matrix. Its elements, for i, j =1, 2,...,n are defined in terms
of the binomial coefficients , so that the i, j element for i ≤ j
is
p
i,j
n
=
(−1)
j−1
.
i−1
C
j−1
(4)
:=
(−1)
j−1
(i − 1)!
(j − 1)!(i − j)!
, for i ≥ j.
The key property which we exploit subsequently is that P
n
is involutary in that
P
n
= P
−1
n
,P
n
P
n
= I
n
. (5)
III. M
ULTINOMIAL COEFFICIENTS
To lead into the derivations of our main results, consider
the polynomials (s − 1)
i
for i =0, 1, 2,...,n for some
nonnegative integer n and scalar s, organized as
(s − 1)
0
s
n
(s − 1)
1
s
n−1
·
·
(s − 1)
n
s
0
= P
n+1
s
n
s
n−1
·
·
s
0
. (6)
Now consider the multinomial (3) for all possible polar
binary sequences {a
1
,a
2
, ··· ,a
n
}. Clearly, the expansion is
invariant of the ordering of the a
i
, so that there are only n +1
selections, namely where there are k =0, 1, 2, ··· ,n values
of a
i
=1, with correspondingly n − k =0, 1, ··· ,n values
of a
i
= −1. Indeed the terms involving sums of products of
the a
i
in (3) are given, for each k =0, 1, 2, ··· ,n,asthe
coefficients of the expansion (s − 1)
k
(s +1)
n−k
.
A. A generalized Pascal matrix
A useful generalization of (6) is then
(s − 1)
0
(s +1)
n
(s − 1)
1
(s +1)
n−1
·
·
(s − 1)
n
(s +1)
0
= R
n+1
s
n
s
n−1
·
·
s
0
(7)
for some readily calculated (n +1)× (n +1) matrix R
n+1
:=
(r
i,j
) consisting of elements r
i,j
, and termed here a general-
ized Pascal matrix. In particular, the i
th
row of R
n+1
consists
of the sums of products of the a
i
in (3), for k values of a
i
=1,
with correspondingly n − k values of a
i
= −1, and are the
coefficients of the polynomial (s − 1)
k
(s +1)
n−k
.
For reference, the cases for n =1, 2 are spelt out as,
R
2
=
11
1 −1
(8)
R
3
=
12 1
10−1
1 −21
(9)
A recursive relationship between the elements of R
k+1
,and
that of R
k
, being a generalization of Pascal’s equations, are
given for k =2, 3, 4, ··· ,n, initialized by (8), as
r
i,1
k+1
:= 1, for i =1, 2, ··· ,k+1;
r
i,j
k+1
:= r
i,j
k
+ r
i,j−1
k
, for j =2, 3, ··· ,k+1;
r
k+1,j
k+1
:= r
k,j
k
− r
k,j−1
k
, for j =2, 3, ··· ,k+1.
(10)
This result is proved in a straightforward manner by induction,
and is not spelt out here.
B. Coefficients via the generalized Pascal matrix
As already noted, the multinomial (3), for each possible
a
1
,a
2
, ··· ,a
n
selection, is invariant of the ordering of the a
i
,
and there are then but n +1 possible multinomials. These can
then be organized as,
s
∗
:=
sign
∗
(n)
sign
∗
(n − 2)
·
·
sign
∗
(n − n)
= R
n+1
ρ
0
ρ
1
·
·
ρ
n
= R
n+1
ρ.
(11)
This relationship means that the desired coefficients are the
solutions of a linear equation as emphasized in the lemma.
Lemma 3.1 The multinomial representation of the sign
∗
function of (3) has coefficients ρ satisfying the linear equations
(11), restated as,
R
n+1
ρ = s
∗
, (12)
where R
n
, the generalized Pascal matrix, is defined recursively
in (8), and (10).
C. Inverse and decomposition of the generalized Pascal matrix
The nature of the inverse of R
n+1
now assumes importance.
We next develop our second main result, namely that R
n
has
a factorization in terms of the Pascal matrix P
n
, and inherits
the involutary property to within a scaling. In particular, we
claim,
Lemma 3.2 The generalized Pascal matrix R
n
, as defined
recursively in (8), and (10), has the scaled involutary property
R
2
n
=2
n−1
I
n
,R
−1
n
=2
1−n
R
n
. (13)
Proof: This result follows by induction arguments. We work
with matrices in lower triangular form. First define F
n
as the
matrix P
n
flipped both left to right and top to bottom. In
obvious notation, we write,
F
n
:= flip(P
n
), or f
i,j
n
= p
n−i,n−j
n
. (14)
Also, define diagonal matrices, in obvious notation, as
D
n
:= diag{2
0
, 2
1
, 2
2
, ··· , 2
n−1
},S
n
:= diag(P
n
). (15)
To proceed with the lemma proof, a decomposition lemma
is now stated and proved,
Lemma 3.3 The generalized Pascal matrix R
n
, as defined
recursively in (8) and (10), has the decomposition in terms of
triangular and diagonal matrices as
R
n
=2
n−1
S
n
F
n
D
−1
n
P
n
= P
n
D
n
F
n
S
n
. (16)
Proof: This lemma result follows by induction, which is
relatively straightforward because only upper or lower tri-
angular matrices are involved. Our approach is guided by
keeping in mind the connection of the matrix elements with
polynomial coefficients. Thus an equivalent result to (16) is to
post-multiply R
n
by the vector [s
n−1
s
n−2
...s
0
]
and apply
both (6) and (7) so that,
(s − 1)
0
(s +1)
n−1
(s − 1)
1
(s +1)
n−2
·
·
(s − 1)
n−1
(s +1)
0
= P
n
2
0
(s +1)
n
2
1
(s +1)
n−1
·
·
2
n−1
(s +1)
0
,
= F
n
(2s)
n−1
(s +1)
0
(2s)
n−2
(s +1)
1
·
·
(2s)
0
(s − 1)
n−1
.
These equations are now in a form that they can be verified
by straightforward induction arguments. The pattern of the
argument becomes clear in passing from n =1to n =2,and
n =2to n =3, so that passing from n to n +1 is then
straightforward. It is necessary to exploit the Pascal equations
which are inherent in the Pascal matrix P
n
construction, and
suitably adjusted for the ‘flipped’ version F
n
. Further details
are omitted.
Proof: (Continuation of Proof for Lemma III.2) The proof
of (13) follows from (16) by substitution and noting in turn
that S
n
,P
n
,F
n
are each readily verified as involutary. Thus,
(R
n
)(R
n
)=(P
n
D
n
F
n
S
n
)(2
n−1
S
n
F
n
D
−1
n
P
n
),
=2
n−1
P
n
D
n
F
n
F
n
D
−1
n
P
n
,
=2
n−1
P
n
D
n
D
−1
n
P
n
,
=2
n−1
P
n
P
n
,
=2
n−1
I
n
.
D. Coefficients from columns of the generalized Pascal matrix
The above Lemmas 3.1, 3.2 together give our main result
stated as a theorem.
Theorem 3.1 The multinomial representation of the sign
∗
function of (3) has coefficients ρ satisfying the linear equations
(11), restated as,
ρ =2
−n
R
n+1
s
∗
. (17)
where R
n
, the generalized Pascal matrix, is defined recursively
in (8), and (10), and satisfies (13) and (16).
This result means that matrix inverses are avoided in calcu-
lating coefficients. This becomes significant for large n.
This result for sign
∗
(sum) functions generalizes trivially
to any nonlinear function f(a
1
,a
2
,...,a
n
) which is invariant
of the ordering of the a
i
.Thes
∗
vector is then replaced by a
vector with j
th
element f(−1, −1,...,1, 1, 1,...,1),where
there are j elements of the data set being −1,andn − j unity
elements.
For completeness, we tabulate the coefficients for low n,
and point out certain properties which can be established by
induction.
n=2 n=3 n=4 n=5 n=6 n=7 n=8
ρ
0
0 0 0 0 0 0 0
ρ
1
1
2
1
2
3
8
3
8
5
16
5
16
35
128
ρ
2
0 0 0 0 0 0 0
ρ
3
0 −
1
2
−
1
8
−
1
8
−
5
80
−
5
80
−
5
128
ρ
4
0 0 0 0 0 0 0
ρ
5
0 0 0
3
8
5
80
5
80
3
128
ρ
6
0 0 0 0 0 0 0
ρ
7
0 0 0 0 0 −
5
16
−
5
128
TABLE I
T
ABLE FOR sign FUNCTION MULTINOMIAL COEFFICIENTS.
n=2 n=3 n=4 n=5 n=6 n=7 n=8
ρ
0
1
2
0
3
8
0
5
16
0
35
128
ρ
1
1
2
1
2
3
8
3
8
5
16
5
16
35
128
ρ
2
−
1
2
0 −
1
8
0 −
5
80
0 −
5
128
ρ
3
0 −
1
2
−
1
8
−
1
8
−
5
80
−
5
80
−
5
128
ρ
4
0 0
3
8
0
5
80
0
3
128
ρ
5
0 0 0
3
8
5
80
5
80
3
128
ρ
6
0 0 0 0 −
5
16
0 −
5
128
ρ
7
0 0 0 0 0 −
5
16
−
5
128
ρ
8
0 0 0 0 0 0
35
128
TABLE II
T
ABLE FOR sign
±
FUNCTION MULTINOMIAL COEFFICIENTS.
Specific relationships between the coefficients are clear from
the tables and can be proved by induction arguments, as
follows. For Table I, for the sign operation,
ρ
(n)
i
=0, for i =0, 1, 3,...and all n,
ρ
(n)
i
= ρ
(n−1)
i
, for all i and n =3, 5, 7,...,
sign(ρ
(n)
i
)=−1, for all n and i =3, 7, 11,...,
sign(ρ
(n)
i
)=1, for all n and i =1, 5, 9 ..., (18)
and for Table II, for the sign
±
operation,
ρ
(n)
i
=0, for i =0, 2, 4,...,
and n =3, 5, 7,...,
ρ
(n)
i
= ρ
(n−1)
i
, for i =1, 3, 5,...,
and n = i +2,i+4,i+6,...,
ρ
(n)
i
= ρ
(n)
i−1
, for i =1, 3, 5,...,
and n = i +1,i+3,i+5,...,
sign(ρ
(n)
i
)=+1, for all n
and i =0, 1, 4, 5, 8, 9 ...,
sign(ρ
(n)
i
)=−1, for all n
and i =2, 3, 6, 7, 10, 11,...,
(19)
There is also symmetry in the coefficients for each odd n.
Indeed for this case the coefficients for sign and sign
±
are
identical (since then sign
±
≡ sign).
We see that Table II can be constructed using these various
properties and the entries in Table I. Moreover, all coefficients
can be constructed from the subset of Table I, namely the ρ
n
i
for i, n odd, i<n/2.
It is readily seen that for n>2, and either coefficient
selection, in obvious notation, then
n+1
i=1
P
n+1
(n, i)ρ
(n)
i−1
=
−1,and
n+1
i=1
P
n+1
(n − 1,i)ρ
(n)
i−1
=0. There are other
products of the rows of P
n
and ρ vectors which are also 0
or 1 not spelt out.
The generalized Pascal Matrix is the key to the coefficients.
It is worth pointing out that although this matrix is not
orthogonal, induction arguments show that all odd rows are
orthogonal to all even rows, so that R
n
R
n
has zero i, j entries
where i is even and j is odd.
IV. C
ONCLUSIONS
Majority logic coding for communication systems has at-
tractive advantages in terms of the simplicity of the decoding.
This is achieved at the expense of optimality. The majority
logic operations involved are highly nonlinear, so there has
been a paucity of theory for developing codes and guaranteeing
properties.
A key step in this direction, presented in this paper, has
been the generation of an explicit formula for the multinomial
representation of the various sign
∗
operations involved in
majority logic. The formula is readily calculated in terms
of binomial coefficients, appearing in a proposed generalized
Pascal matrix. A factorization of this matrix, in terms of a
lower Cholesky factor of the original Pascal matrix, turns out
to simplify the proof and derivation of the coefficients. The
results are more complete than hitherto given for the case of
sign, and are new for the sign
±
case.
V. A
CKNOWLEDGEMENTS
Thanks to GO-CDMA Ltd. of Hong Kong SAR, China, for
the use of their intellectual properties in this work.
John’s work is supported in part by the ARC discovery
grants A0010582, DP0450539, and in part by the National ICT
Australia (NICTA), which is funded by the Australian Gov-
ernment Backing Australia’s Ability Initiative in part through
the Australian Research Council.
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