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Properties of Binary Pearson Codes

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We consider the transmission and storage of data that use coded symbols over a channel, where a Pearson-distance based detector is used for achieving resilience against unknown channel gain and offset, and corruption with additive noise. We discuss properties of binary Pearson codes, such as the Pearson noise distance that plays a key role in the error performance of Pearson-distance-based detection. We also compare the Pearson noise distance to the well-known Hamming distance, since the latter plays a similar role in the error performance of Euclidean distance- based detection.
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Properties of Binary Pearson Codes
Jos H. Weber
Delft University of Technology
Delft, The Netherlands
Email: j.h.weber@tudelft.nl
Kees A. Schouhamer Immink
Turing Machines Inc.
Rotterdam, The Netherlands
Email: immink@turing-machines.com
Abstract—We consider the transmission and storage of data
that use coded symbols over a channel, where a Pearson-distance-
based detector is used for achieving resilience against unknown
channel gain and offset, and corruption with additive noise. We
discuss properties of binary Pearson codes, such as the Pearson
noise distance that plays a key role in the error performance of
Pearson-distance-based detection. We also compare the Pearson
noise distance to the well-known Hamming distance, since the
latter plays a similar role in the error performance of Euclidean-
distance-based detection.
I. INTRODUCTION
In mass data storage devices, the user data are translated
into physical features that can be either electronic, magnetic,
optical, or of other nature. Due to process variations, the
magnitude of the physical effect may deviate from the nominal
values, which may effect the reliable read-out of the data. For
example, over a long time, the charge in memory cells, which
represents the stored data, may fade away, and as a result
the main physical parameters change resulting in channel
mismatch and increased error rate. It has been found that such
retention errors to be dominant errors in solid-state memories.
The detector’s ignorance of the exact value of the channel’s
main physical parameters [1], [2], [3], [4], a phenomenon
called channel mismatch, may seriously degrade the error per-
formance of a storage or transmission medium [5]. Researchers
have been seeking for methods that can withstand channel
mismatch. Immink and Weber [5] advocated a novel data
detection method based on the Pearson distance that offers
invariance, “immunity”, to offset and gain mismatch. They
also showed the other side of the medal of Pearson-distance-
based detection, namely that it is less resilient to additive noise
than conventional Euclidean-distance-based detection.
In this paper, we investigate the relationship between the
noise resilience of Euclidean distance versus Pearson distance
detection. The outline of the paper is as follows. In Section II,
we present the channel model under consideration, we re-
capitulate the relevant prior art of minimum Euclidean and
Pearson distance detection, and we review the definition of
Pearson codes. In Section III, we discuss properties of binary
Pearson codes, such as lower and upper bounds to the error
performance of detectors based on the Pearson distance, and
the difference in noise resilience of minimum Euclidean versus
minimum Pearson distance detection. Section IV concludes
our paper.
II. BACKGROUND AND PRELIMINARIES
In this section we present some prior art, mainly from [5],
and set the scene for the results of this paper.
A. Pearson distance
We start with the definition of two quantities of an n-vector
of reals, z, namely the average of zby z=1
nn
i=1 ziand
the (unnormalized) variance of zby
σ2
z=
n
i=1
(ziz)2.(1)
The Pearson distance between the vectors xand yin Rnis
defined by
δp(x,y) = 1 ρx,y,(2)
where the (Pearson) correlation coefficient [6] is defined by
ρx,y=n
i=1(xix)(yiy)
σxσy
.(3)
It is immediate that the Pearson distance δp(x,y)is undefined
if xor yhas variance zero, i.e., is a ‘constant’ vector
(c, c, . . . , c)with cR. Further, note that the Pearson distance
is not a regular metric, but a measure of similarity between
the vectors xand y. It can easily be verified that the triangle
inequality condition, δp(x,z)δp(x,y) + δp(y,z), is not
always satisfied. For example. let n= 4 and let x= (0001),
y= (0011), and z= (0010), then δp(x,z) = 1.3333,
δp(x,y) = 0.4226, and δp(y,z) = 0.4226.
B. Channel model
We assume a simple linear channel model where the sent
codeword c, taken from a finite codebook S ⊂ Rn, is received
as the real-valued vector
r=a(c+ν) + b1,(4)
where 1is the all-one vector (1,1,...,1) of length n, while
a,a > 0, is an unknown gain, bRis an unknown offset,
and ν=(ν1, . . . , νn)is additive noise with νiRbeing
zero-mean independent and identically distributed (i.i.d) noise
samples with Gaussian distribution N(0, σ2), where σ2R
denotes the variance. We assume that the parameters aand b
vary slowly, so that during the transmission of the nsymbols
in a codeword the parameters aand bare fixed, but that these
values may be different for the next transmitted codeword.
C. Detection
Aminimum Pearson distance detector outputs a codeword
according to the minimum distance decision rule
cp= arg min
ˆ
c∈S
δp(r,ˆ
c).(5)
Due to the properties of the Pearson correlation coefficient
such a detector is immune to gain and offset mismatch [5].
However, it is more sensitive to noise than the well-known
minimum Euclidean distance detector which outputs
ce= arg min
ˆ
c∈S
δ2(r,ˆ
c),(6)
with
δ2(x,y) =
n
i=1
(xiyi)2(7)
being the squared Euclidean distance between xand y.
The computation of the probability that a minimum Pearson
distance detector errs has been investigated in [5]. A principal
finding is that it is not the (minimum) Pearson distance,
δp(x,y), between codewords xand y, that governs the error
probability, but a quantity called Pearson noise distance, which
is denoted by d(x,y). The squared Pearson noise distance,
d2(x,y), between the vectors xand yis given by
d2(x,y) = 2σ2
xδp(x,y) = 2σ2
x(1 ρx,y).(8)
The union bound estimate of the word error rate (WER) is
WERPear 1
|S|
x∈S
y∈S,x̸=y
Qd(x,y)
2σ,(9)
where Q(x) = 1
2π
xeu2
2du is the well-known Q-
function. Note the similarity with the union bound for the
WER in case of a Euclidean detector, which reads
WEREucl 1
|S|
x∈S
y∈S,x̸=y
Qδ(x,y)
2σ(10)
for an additive Gaussian noise channel, i.e., a channel as in
(4) with the values of aand bbeing known to the receiver.
We emphasize again that for WER performance of the Pearson
distance detector it does not matter whether the gain and offset
values are known to the receiver or not, while the performance
of the Euclidean distance detector quickly deteriorates when
the gain and offset drift away from their ideal values a= 1
and b= 0 while being unknown to the receiver [5].
For small σ, the WER is dominated by the term with the
smallest distance between any xand any different y∈ S, so
that
WERPear Np,minQdmin
2σ, σ 1,(11)
where dmin = minx,y∈S,x̸=yd(x,y)and Np,min is the num-
ber of codewords y(called nearest neighbors) at minimum
Pearson noise distance dmin from x, averaged over all x∈ S.
D. Pearson codes
In order to allow easy encoding and decoding operations,
it is common to use a q-ary codebook S, i.e., S⊆ Qnwith
Q={0,1, . . . , q 1}. Since a minimum Pearson distance
detector cannot deal with codewords cwith σc= 0 and cannot
distinguish between the words cand c11+c2c,c2>0,
well-chosen words must be barred from Qnto guarantee
unambiguous detection. Weber et al. [7] coined the name
Pearson code for a set of codewords that can be uniquely
decoded by a minimum Pearson distance detector. Codewords
in a Pearson code Ssatisfy two conditions, namely
Property A: If c∈ S then c11+c2c/∈ S for all c1, c2R
with (c1, c2)̸= (0,1) and c2>0;
Property B: c1/∈ S for all cR.
For a binary Pearson code, i.e., q= 2, this implies that only
two vectors must be barred, namely the all-‘0’ vector 0and
all-‘1’ vector 1. Hence, the largest binary Pearson code of
length nis
Pn={0,1}n\ {0,1}.(12)
However, in order to improve the error performance, it may
be necessary to further restrict the codebook, particularly by
avoiding codeword pairs with a small Pearson noise distance.
In the next section we investigate properties of the Pearson
(noise) distance and detector that provide more insight and as
such could be useful in the process of designing good Pearson
codes.
III. PROP ERT IE S OF B INARY PEARSON CODES
In this section, we study the important binary case, q= 2.
Particularly, we will determine bounds on the Pearson noise
distance and make comparisons with the Hamming distance.
First we give some notation. Let xand ybe two n-vectors
taken from the code S⊂ {0,1}n. We define the integers
wx=
n
i=1
xi, wy=
n
i=1
yi, wxy =
n
i=1
xiyi,(13)
where wxand wyare the weights of the vectors xand y,
respectively, and wxy, the index of ‘1’-coincidence (or overlap)
of the vectors xand y, denotes the number of indices iwhere
xi=yi= 1. Note that all additions and multiplications in
(13) are over the real numbers.
For clerical convenience, we define the real-valued function
φn(wx, wy, wxy) = d(x,y).(14)
Using (8) and the above definitions, we have
φ2
n(wx, wy, wxy) = 2σ2
x1wxy wxwy
n
σxσy,(15)
where
σ2
x=wxw2
x
nand σ2
y=wyw2
y
n.(16)
For all x,y∈ Pn, the integer variables wx,wy, and wxy
satisfy
1wx, wyn1,(17)
max{wx+wyn, 0} ≤ wxy min{wx, wy},and (18)
wxy wx1if x̸=yand wx=wy.(19)
In the next subsections we present the main results of this
paper.
A. Bounds on the Pearson noise distance
Since the Pearson noise distance d(x,y)plays a crucial role
in the performance of a Pearson code, we should investigate
which values it can take. We start with a simple upper bound.
Theorem 1: For any two codewords xand yin Pn,n2,
it holds that
d2(x,y)4σ2
xnif nis even,
n1/n if nis odd,
where equality holds in the first inequality if and only if y=
1x, while equality holds in the second inequality if and
only if wx=n/2or wx=n/2.
Proof. It is a well-known property of the Pearson correlation
coefficient, ρu,v, of any two real-valued non-constant vectors
uand vof the same length, that |ρu,v| ≤ 1and also that
ρu,v=1if and only if v=c11+c2u, where the coefficients
c1and c2,c2<0, are real numbers [6, Sec. IV.4.6]. Hence,
for any x∈ Pn,d2(x,y)is maximized over all y∈ Pnif
and only if y=1x, i.e., by setting yas the inverse of
x. The results as stated in the theorem now easily follow by
observing that
d2(x,1x) = φ2
n(wx, n wx,0) = 4σ2
x= 4 wxw2
x
n
and that the last expression is maximized if and only if wx=
n
2or wx=n
2.
In case two codewords have equal weight, we have the
following useful observation.
Lemma 1: For any two codewords xand yin Pn,n2,
of equal weight, it holds that
d2(x,y) = 2(wxwxy).
Proof. From (14)-(16) and the fact wx=wyit follows that
d2(x,y) = 2σ2
x1wxy w2
x
n
σ2
x= 2 σ2
xwxy +w2
x
n
= 2(σ2
xwxy +wxσ2
x) = 2(wxwxy),
which shows the stated result.
The minimum Pearson noise distance, dmin, between any
two different codewords plays a key role in the evaluation
of the error performance of the minimum Pearson detector,
see (11). The next theorem shows that dmin of Pnequals
φn(1,2,1). This was already conjectured in [5], but is now
formally proved.
Theorem 2: For any two different codewords xand yin
Pn,n3, it holds that
d2(x,y)φ2
n(1,2,1) = 2n2
n1n2
2n2,
where equality holds if and only if wx=wxy = 1,wy= 2
or wx=n1,wy=wxy =n2.
Proof. Our strategy is to look for three integers, wx,wy, and
wxy, that minimize the function φn(wx, wy, wxy ), under the
constraints (17)-(19). Any two different codewords xand y
having the found parameters will then minimize d(x,y). Since
ρx,y=ρy,x, it follows from (8) that it holds for such xand
ythat σ2
xσ2
y, i.e., wxwynwx. Further, we may
and will assume wxn/2since
d(x,y) = d(1x,1y)(20)
for all xand yin Pn.
With regard to the selection of the integer wxy, it is
straightforward from (15) that we should choose it as large as
possible for any values of wxand wy. We distinguish between
the cases wx=wyand wx< wy.
In case wx=wy, the value of wxy is at most wx1
since x̸=y. Hence, from Lemma 1, we find d2(x,y) =
2(wxwxy)2. Note that the expression in the theorem is
clearly smaller than 2.
In case wx< wy, the maximum value of wxy is wx. Note
that wx< wyimplies that 1wx≤ ⌊(n1)/2. We proceed
with the selection of wy. From (14)-(16), we have
φ2
n(wx, wy, wx) = 2σ2
x(1 α),(21)
where
α2=wxwxwy
n2
σ2
xσ2
y
=1
wy1
nw2
x
σ2
x
, α > 0.(22)
It is immediate from (21) and (22) that, for any value of
wx, the function φn(wx, wy, wx)is at a minimum when the
factor 1
wy1
nis at a maximum. We conclude that, for all wx,
the choice wy=wx+ 1 minimizes (21). Subsequently, we
substitute wy=wx+ 1, and analyze the function
ψn(wx) = φ2
n(wx, wx+ 1, wx) = 2σ2
x(1 β)(23)
in the single (integer) variable, wx, where, using (22), we write
β2=1
wx+ 1 1
nw2
x
σ2
x
=wx(n1wx)
(wx+ 1)(nwx), β > 0.
(24)
In order to determine the value of wx∈ {1,2, . . . , (n1)/2⌋}
minimizing ψn(wx), we consider the function fn(w)which is
obtained by replacing the discrete variable wxin ψn(wx)by
the continuous variable w, with w[1,(n1)/2]. We
replace wxby win (24) as well and then express win β,
obtaining
w=n1
2n
2gn(β),(25)
where
gn(β) = 1 + 1
n2+2(β2+ 1)
n(β21),0< β0ββ1<1,
β2
0=β2|w=1 =n2
2n2,
0 5 10 15 20 25 30
0.76
0.77
0.78
0.79
0.8
0.81
0.82
0.83
n
minimum Pearson noise distance
Fig. 1. Minimum Pearson noise distance of Pn.
and
β2
1=β2|w=n1
2=n1
2⌋⌈n1
2
n+1
2⌋⌈n+1
2.
Note that gn(β)is strictly decreasing with βon the interval
under consideration, and thus wis a strictly increasing function
of β. Next, we substitute (25) in fn(w), and the resulting
function with variable βis
hn(β) = n21
2nng(β)
2g(β)(1 β)
=β1
n+β2+ 1
β+ 1 + (β1)g(β).(26)
It is not hard to show that the three terms in (26) are all strictly
increasing with βin the range β0ββ1, so that hn(β)is
at a minimum for β=β0. Thus, φn(wx, wy, wx)achieves a
minimum when wx=wxy = 1 and wy= 2. The expression
stated in the theorem follows by substituting these parameters
into (15). Because of (20), also the choice wx=n1and
wy=wxy =n2achieves this minimum. Finally, it follows
from the strict monotonicity of the functions used in the above
derivation that no other choices achieve the minimum Pearson
noise distance.
Note that it follows from this theorem that, for large values
of n, the minimum Pearson noise distance of the code Pn
approaches 220.765. A graphical representation is
provided in Figure 1.
We conclude this subsection with a look at the number
of codeword pairs having a certain Pearson noise distance
between each other. In this respect, note that the number of
pairs (x,y)with given values for wx,wy, and wxy is
n!
wxy!(wxwxy )!(wywxy )!(nwxwy+wxy )!,(27)
which easily follows from standard combinatorial arguments.
For example, it follows from this result and Theorem 2 that the
number of codeword pairs (x,y)in Pnat minimum Pearson
noise distance is 2×n!
1!0!1!(n2)! = 2n(n1). Hence, dividing
this expression by the number of codewords gives Np,min,
which can be used, together with the minimum distance result
from Theorem 2, in (11) to obtain an approximate value for
the WER of a Pearson distance based detector.
B. Hamming versus squared Pearson noise distance
The Hamming distance between two vectors is an essen-
tial notion in coding theory, and a comparison between the
properties of Hamming and Pearson distance is therefore
relevant. Since xi, yi∈ {0,1}, the Hamming distance equals
the squared Euclidean distance, i.e.,
dH(x,y) =
n
i=1
(xiyi)2=wx+wy2wxy.(28)
It is essential that we define a fair yardstick for quantify-
ing the noise resilience of minimum Euclidean and Pearson
distance detection. To that end, we consider the ratio between
the squared Pearson noise distance and the Hamming distance,
denoted by gx,y, i.e.,
gx,y=d2(x,y)
dH(x,y).(29)
It follows from the WER analysis in Subsection II-C that this
ratio being smaller than one implies that the Euclidean detector
is more resilient to noise than the Pearson detector in case xis
transmitted and yis considered as an alternative for xin the
decoding process. Vice versa for this ratio being larger than
one.
As a first observation, note that it follows from Lemma 1
and (28) that d2(x,y) = dH(x,y)and thus gx,y= 1 in
case xand yare of equal weight. Evidently, there is no
error performance difference between minimum Pearson and
Euclidean detectors for codewords drawn from a constant
weight set.
In the remainder of this subsection, we consider vectors x
and yfrom Pnwith the weight of xbeing fixed at wx
{1,2, . . . , n 1}and the Hamming distance dH(x,y)being
fixed at dH∈ {1,2, . . . , n}. Since the overlap wxy of xand
yis expected to have a high impact on gx,y, we consider
two extreme options for wxy in our analysis: 1) we choose
wy∈ {1,2, . . . , n 1}such that wxy is as small as possible,
2) we choose wysuch that wxy is as large as possible, in both
cases under the constraints of the fixed values for the weight
of xand the Hamming distance between xand y.
Case 1: It follows in a straightforward way that the minimal
overlap of xand yis
wxy =
wxdHif 1dHwx1,
1if dH=wx,
0if wx+ 1 dHn,
(30)
achieved for
wy=
wxdHif 1dHwx1,
2if dH=wx,
dHwxif wx+ 1 dHn.
(31)
0 2 4 6 8 10 12 14 16 18 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Hamming distance
g
n=20, w x=6, wxy minimal
g
y,x
g
x,y
Fig. 2. The ratios gx,yand gy,xin case n= 20,wx= 6,1dH
20, and wy∈ {1,2,...,19}chosen such that wxy is minimized
(Case 1).
0 2 4 6 8 10 12 14 16 18 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Hamming distance
g
n=20, w x=6, wxy maximal
g
y,x
g
x,y
Fig. 3. The ratios gx,yand gy,xin case n= 20,wx= 6,1dH
20, and wy∈ {1,2, . . . , 19}chosen such that wxy is maximized
(Case 2).
Case 2: Similarly, we have that the maximal overlap of x
and yis
wxy =
wxif 1dHnwx1,
wx1if dH=nwx,
ndHif nwx+ 1 dHn,
(32)
achieved for
wy=
wx+dHif 1dHnwx1,
n2if dH=nwx,
2ndHwxif nwx+ 1 dHn.
(33)
The g-ratios can now be obtained from (29) by applying
(14), (15), and (28). Figures 2 and 3 show, for Cases 1 and 2,
respectively, the resulting gx,yand gy,xvalues for n= 20,
wx= 6, and 1dH20.
Several interesting observations can be made from these
figures. First of all, note that there are ‘irregularities’ for the
gy,xcurves at dH=wx= 6 (Case 1) and dH=nwx= 14
(Case 2). For Case 1 this can be explained as follows. From
(30) we see that wxy equals max{0, wxdH}, except when
dH=wx= 6, because this would imply wy= 0 (impossible
since 0/∈ Pn). Hence, wxy = 1 >0for dH= 6, which leads
to the observed notch. Similarly for Case 2 (using 1/∈ Pn).
Further, we observe that all curves end at the same point.
This is due to the fact that for dH=nthe only possible
options for wyand wxy when wxis given read wy=nwx
and wxy = 0. The resulting value is
gx,1x=g1x,x=4σ2
x
n=4wx1wx
n
n(34)
in general, and thus 0.84 for the example under consideration.
Finally, note that, as expected, the largest g-ratios are found
in Case 1. Most strikingly, we see that these ratios may even
exceed the value one (see Figure 2), suggesting that the noise
resistance of the Pearson detector is higher than the noise
resistance of the Euclidean detector for these cases. Of course,
this cannot be true, since a Euclidean detector is well-known
to be optimal in case of Gaussian noise. Indeed, we observe
that in all cases that gx,yexceeds one, its counterpart gy,x
is smaller than one. Similarly, gy,x>1implies gx,y<1.
Since codeword pairs with smaller distances are dominant
with respect to contributions to the WER, the overall result
is still that from the noise perspective Euclidean detectors
are superior to Pearson detectors, which is the price to be
paid for the immunity of the latter detectors to gain and
offset mismatches. The analysis as done in this paper can be
exploited in the design of new Pearson codes, i.e., subsets
of Pn, with a noise performance closer to the Euclidean
case, by avoiding the selection of codeword pairs with small
Pearson noise distances. It is clear that in order to increase
the Pearson noise distance, the focus should not only be on
Hamming distance increase, since these two distance measures
are certainly not growing proportionally. Rather, also the
codeword weights must be taken into account.
IV. CONCLUSIONS
We have investigated various properties of Pearson-distance-
based detection and Pearson codes. For binary codes, we have
derived upper and lower bounds on the Pearson noise distance
and studied relations with the Hamming distance.
As possibilities for future work we identify (i) application
of the findings in order to construct codes with an increased
minimum Pearson noise distance and (ii) extension of the
results to q-ary codes.
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... The use of the Pearson distance requires that the set of codewords satisfies several specific properties. Such sets are called Pearson codes, which have attracted a lot of interest [51][52][53][54][55]. In [51], optimal Pearson codes are presented, in the sense of having the largest number of codewords and thus minimum redundancy among all q-ary Pearson codes of fixed length n. ...
... In [51], optimal Pearson codes are presented, in the sense of having the largest number of codewords and thus minimum redundancy among all q-ary Pearson codes of fixed length n. Properties of binary Pearson codes are discussed in [52,53], where the Pearson noise distance is compared to the well-known Hamming distance. A simple systematic Pearson coding scheme, that maps sequences of information symbols generated by a q-ary source to q-ary code sequences, is proposed in [54]. ...
... Like the offset b, the gain a is assumed to be constant for a transmitted codeword, but it may vary from one codeword to the next. A decoding criterion, that is immune to both gain and offset mismatch, has been proposed in [8], and some basic properties for the binary case have been presented in [13]. An interesting topic for future research is the design of suitable codes for this scenario as well. ...
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Decoders minimizing the Euclidean distance between the received word and the candidate codewords are known to be optimal for channels suffering from Gaussian noise. However, when the stored or transmitted signals are also corrupted by an unknown offset, other decoders may perform better. In particular, applying the Euclidean distance on normalized words makes the decoding result independent of the offset. The use of this distance measure calls for alternative code design criteria in order to get good performance in the presence of both noise and offset. In this context, various adapted versions of classical binary block codes are proposed, such as (i) cosets of linear codes, (ii) (unions of) constant weight codes, and (iii) unordered codes. It is shown that considerable performance improvements can be achieved, particularly when the offset is large compared to the noise.
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