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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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https://ro.ecu.edu.au/ecuworks/2757

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.

R

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