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Simple Systematic Pearson Coding

Jos H. Weber∗,∗∗

∗Delft University of Technology

The Netherlands

j.h.weber@tudelft.nl

Theo G. Swart∗∗

∗∗University of Johannesburg

South Africa

tgswart@uj.ac.za

Kees A. Schouhamer Immink∗∗∗

∗∗∗Turing Machines Inc.

The Netherlands

immink@turing-machines.com

Abstract—The recently proposed Pearson codes offer immunity

against channel gain and offset mismatch. These codes have very

low redundancy, but efﬁcient coding procedures were lacking. In

this paper, systematic Pearson coding schemes are presented. The

redundancy of these schemes is analyzed for memoryless uniform

sources. It is concluded that simple coding can be established at

only a modest rate loss.

I. INT ROD UC TI ON

Dealing with rapidly varying offset and/or gain is an im-

portant issue in signal processing for modern storage and

communication systems. For example, methods to solve these

difﬁculties in Flash memories have been discussed in, e.g., [7],

[9], and [11]. Also, in optical disc media, the retrieved signal

depends on the dimensions of the written features and upon

the quality of the light path, which may be obscured by

ﬁngerprints or scratches on the substrate, leading to offset

and gain variations of the retrieved signal. Automatic gain

and offset control in combination with dc-balanced codes

are applied albeit at the cost of redundancy [4], and thus

improvements to the art are welcome.

Immink and Weber [5] showed that detectors that use the

Pearson distance offer immunity to offset and gain mismatch.

Use of the Pearson distance demands that the set of codewords

satisﬁes certain special properties. Such sets are called Pearson

codes. In [10], optimal codes were 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.

However, the important issue of efﬁcient coding procedures

was not addressed. In this paper, we present simple systematic

Pearson coding schemes, mapping sequences of information

symbols generated by a q-ary source to q-ary code sequences.

The redundancy of these coding schemes is analyzed for

memoryless sources generating q-ary symbols with equal

probability.

The remainder of this paper is organized as follows. In

Section II, we review the concepts of Pearson detection and

q-ary Pearson codes. Then, in Section III, we present our

systematic coding schemes and analyze their redundancy.

Finally, in Section IV, we draw conclusions.

II. PR EL IM INA RI ES

A. Codes and Redundancies

Let Cbe a q-ary code of length n, i.e., C ⊆ Qn, where

Q={0,1, . . . , q −1}is the code alphabet of size q≥2. Here

the alphabet symbols are to be treated as being real numbers

rather than elements of Zq. The cardinality of the code is

denoted by M, i.e., M=|C|. Usually, the redundancy of

code Cis then deﬁned as

n−logqM. (1)

Actually, this assumes that all codewords are equally likely to

be selected. In a more general setting, an arbitrary probability

mass function (PMF) is speciﬁed on the codewords. Let the

probability that codeword xi∈ C,1≤i≤M, is selected for

transmission or storage be Pi. Since the average amount of

information carried by a codeword is then −M

i=1 PilogqPi

symbols, the redundancy of code Cwith PMF {Pi}is

n+

M

i=1

PilogqPi.(2)

In case Pi= 1/M for all i, then (2) reduces to (1).

B. Pearson Detection

For convenience, we use the shorthand notation av+b

=(av1+b, av2+b, . . . , avn+b). A common assumption

is that a transmitted codeword xis received as a vector

r=a(x+ν) + bin Rn. Here aand bare unknown real

numbers with apositive, called the gain and the (dc-)offset,

respectively. Moreover, νis an additive noise vector, where

the νi∈Rare noise samples from a zero-mean Gaussian

distribution. Note that both gain and offset do not vary from

symbol to symbol, but are the same for the whole block of n

symbols. The receiver’s ignorance of the channel’s momentary

gain and offset may lead to massive performance degradation

as shown, for example, in [5] when a traditional detector,

based on thresholds or the Euclidean distance, is used. In the

prior art, various methods have been proposed to overcome

this difﬁculty. In a ﬁrst method, data reference, or ‘training’,

patterns are multiplexed with the user data in order to ‘teach’

the data detection circuitry the momentary values of the

channel’s characteristics such as impulse response, gain, and

offset. In a channel with unknown gain and offset, we may

use two reference symbol values, where in each codeword,

a ﬁrst symbol is set equal to the lowest signal level and a

second symbol equal to the highest signal level. The positions

and amplitudes of the two reference symbols are known to

the receiver. The receiver can straightforwardly measure the

amplitude of the retrieved reference symbols, and normalize

the amplitudes of the remaining symbols of the retrieved

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codeword before applying detection. Clearly, the redundancy

of the method is two symbols per codeword.

In a second prior art method, codes satisfying equal balance

and energy constraints [2], which are immune to gain and

offset mismatch, have been advocated. However, these codes

suffer from a rather high redundancy. In a recent contribution,

Pearson distance detection is advocated since its redundancy

is much less than that of balanced codes [5]. The Pearson

distance between the vectors uand vis deﬁned as follows. For

a vector u, deﬁne u=1

nn

i=1 uiand σ2

u=n

i=1(ui−u)2.

Note that σuis closely related to, but not the same as, the

standard deviation of u. The (Pearson) correlation coefﬁcient

of uand vis deﬁned by

ρu,v=n

i=1(ui−u)(vi−v)

σuσv

,(3)

and the Pearson distance between uand vis given by

δ(u,v) = 1 −ρu,v.(4)

The Pearson distance and Pearson correlation coefﬁcient are

well-known concepts in statistics and cluster analysis. Since

|ρu,v| ≤ 1, it holds that 0≤δ(u,v)≤2. The Pearson

distance is translation and scale invariant, that is, δ(u,v) =

δ(u, av+b), for any real numbers aand bwith a > 0.

Upon receipt of a vector r, a minimum Pearson distance

detector outputs the codeword arg minx∈C δ(r,x). Since the

Pearson distance is translation and scale invariant, we conclude

that the Pearson distance between the received vector and

a codeword is independent of the channel’s gain or offset

mismatch, so that, as a result, the error performance of the

minimum Pearson distance detector is immune to gain and

offset mismatch, which is a big advantage in comparison to

Euclidean distance detectors. However, Pearson distance de-

tectors are more sensitive to noise. Therefore, hybrid minimum

Pearson and Euclidean distance detectors have been proposed

[6] to deal with channels suffering from both signiﬁcant noise

and gain/offset.

C. Pearson Codes

Its immunity to gain and offset mismatch implies that

the minimum Pearson distance detector cannot be used in

conjunction with arbitrary codes, since δ(r,x) = δ(r,y)if

y=c1+c2x, with c1, c2∈Rand c2positive. In other words,

since a minimum Pearson detector cannot distinguish between

the words xand y=c1+c2x, the codewords must be taken

from a code C ⊆ Qnthat guarantees unambiguous detection

with the Pearson distance metric (4) accordingly. Furthermore,

note that codewords of the format x= (c, c, . . . , c)should not

be used in order to avoid that σx= 0, which would lead to

an undeﬁned Pearson correlation coefﬁcient. In conclusion, the

following condition must be satisﬁed:

If x∈ C then c1+c2x/∈ C for all c1, c2∈R

with (c1, c2)̸= (0,1) and c2≥0. (5)

A code satisfying (5) is called a Pearson code [10]. Known

constructions of Pearson codes read as follows.

•The set of all q-ary sequences of length nhaving at least

one symbol ‘0’ and at least one symbol ‘1’. We denote

this code by T(n, q). It is a member of the class of T-

constrained codes [3], consisting of sequences in which

Tpre-determined reference symbols each appear at least

once.

•The set of all q-ary sequences of length nhaving at least

one symbol ‘0’, at least one symbol not equal to ‘0’,

and having the greatest common divisor of the sequence

symbols equal to ‘1’. We denote this code by P(n, q). It

is has been shown in [10] that this code is optimal in the

sense that it has the largest number of codewords among

all q-ary Pearson codes of length n.

Another code which is of interest, though not being a Pearson

code, is deﬁned as follows.

•The set of all q-ary sequences of length nhaving at least

one symbol ‘0’. We denote this code by Z(n, q). It is also

a member of the class of T-constrained codes [3]. Due

to the presence of the reference symbol ‘0’ it is resistant

against offset mismatch.

Note that

T(n, q)⊆ P(n, q)⊆ Z (n, q).(6)

The cardinalities and redundancies (in the sense of (1)) of

these three codes, as derived in [10], are given in Table I,

where, for a positive integer d, the M¨

obius function µ(d)is

deﬁned [1, Chapter XVI] to be 0if dis divisible by the square

of a prime, otherwise µ(d) = (−1)kwhere kis the number

of (distinct) prime divisors of d.

III. SYS TE MATI C COD IN G

As stated, the Pearson code P(n, q)is optimal in the sense

of having largest cardinality and thus smallest redundancy.

However, an easy coding procedure mapping information

sequences to code sequences and vice versa is not evident

at all. In this section, we propose easy coding procedures,

possibly at the expense of a somewhat higher redundancy.

We only use code sequences of a ﬁxed length n, but for the

information we consider both ﬁxed-length and variable-length

sequences. Hence, ﬁxed-to-ﬁxed (FF) as well as variable-

to-ﬁxed (VF) length coding schemes are proposed. For the

source we make the common assumption that it is memoryless

and that all qsource symbols appear with equal probability

1/q. We start by introducing simple coding schemes resistant

against offset mismatch only. Then we continue with similar

procedures for Pearson coding.

A. Systematic Coding for Z(n, q)

The code Z(n, q)consists of all q-ary sequence of length

ncontaining at least one symbol ‘0’. Its cardinality and

redundancy are given in Table I. Here, we propose simple

coding procedures systematically mapping q-ary information

symbols to code sequences x= (x1, x2, . . . , xn)in Z(n, q).

A well-known extremely simple FF-scheme, which we call

ZFF(n, q), is to ﬁll the code sequence xwith n−1information

symbols in the subsequence (x1, x2, . . . , xn−1)and to set

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TABLE I

CARDINALITY AND REDU NDA NCY O F TH E COD ES T(n, q),P(n, q ),AND Z(n, q).

Cardinality Redundancy

T(n, q)qn−2(q−1)n+ (q−2)n−logq(1−2(q−1

q)n+(q−2

q)n)

≈(2(q−1

q)n−(q−2

q)n)/ln(q)

P(n, q)∑q−1

d=1 µ(d)((⌊q−1

d⌋+ 1)n−⌊q−1

d⌋n−1)−logq(1−(q−1

q)n+O((q+1

2q)n))

=qn−(q−1)n+O(⌈q/2⌉n)as n→ ∞ ≈ ((q−1

q)n+O((q+1

2q)n))/ln(q)

Z(n, q)qn−(q−1)n−logq(1−(q−1

q)n)

≈(q−1

q)n/ln(q)

xn= 0. Due to the ﬁxed last symbol, which acts as a

reference, the redundancy of this method is 1.

Note that while the redundancy of Z(n, q)is decreasing in

n, the redundancy of ZFF(n, q)remains 1. Next, we propose a

systematic VF-scheme, ZVF(n, q), for which the redundancy

decreases in n:

1) Take n−1information from the q-ary source and set

these as (x1, x2, . . . , xn−1).

2) If xi= 0 for at least one 1≤i≤n−1, then choose xn

to be a (new) information symbol, otherwise set xn= 0.

It can easily be seen that the code sequence xis indeed in

Z(n, q)and that the information symbols can be uniquely

retrieved from xby checking whether it contains a zero in

its ﬁrst n−1positions: if ‘yes’, then all ncode symbols

are information symbols, if ‘no’, then only the ﬁrst n−1

code symbols are information symbols. Since the number of

information symbols may vary from codeword to codeword

(being either nor n−1), while the length of the codewords

is ﬁxed at n, this can be considered a variable-to-ﬁxed length

coding procedure. All words in Z(n, q)can appear as code

sequence, but not necessarily with equal probability. This leads

to a redundancy as stated in the next theorem.

Theorem 1. For a memoryless uniform q-ary source, the

redundancy of coding scheme ZVF(n, q)is (1 −1/q)n−1.

Proof: This result can be obtained using (2), with the

observations that (i) Pi= (1/q)n−1for the (q−1)n−1code

sequences xiwith no zeroes among the ﬁrst n−1symbols and

thus with last code symbol equal to zero, and (ii) Pi= (1/q)n

for the other q(qn−1−(q−1)n−1)code sequences xiwith

at least one zero among the ﬁrst n−1symbols. Hence, the

resulting redundancy is

n+

M

i=1

PilogqPi

=n+ (q−1)n−1(1/q)n−1logq(1/q)n−1+

q(qn−1−(q−1)n−1)(1/q)nlogq(1/q)n

= (1 −1/q)n−1.

Another way to derive this result is to observe that the

TABLE II

ZVF(3,2) C ODI NG F OR A ME MO RYLE SS U NIF OR M BIN ARY SO URC E.

Info Codeword ∈ Z(3,2) Probability Redundancy

000 000 1/8 0

001 001 1/8 0

010 010 1/8 0

011 011 1/8 0

100 100 1/8 0

101 101 1/8 0

11 110 1/4 1

probability of the case that a sequence of n−1information

symbols does not contain a zero, leading to one redundant

symbol, is equal to (1 −1/q)n−1, while the opposite case

leads to no redundancy at all. The weighted average

(1 −1/q)n−1×1 + (1 −(1 −1/q)n−1)×0 = (1 −1/q)n−1

then gives the redundancy of ZVF(n, q).

As an example, we consider scheme ZVF(3,2) for a mem-

oryless binary source producing zeroes and ones with equal

probability. The seven codewords of Z(3,2) are then used

with probabilities as indicated in Table II, and thus the average

redundancy is 1/4. This result can be obtained by applying (2),

i.e., 3+6×(1/8) log2(1/8) + (1/4) log2(1/4) = 1/4, or by

directly applying Theorem 1, i.e, (1 −1/2)2= 1/4. Note that

achieving the somewhat lower redundancy 3−log2(7) = 0.19

of the code Z(3,2) as such would require all seven codewords

to be used with probability 1/7, which does not naturally match

the source statistics.

In conclusion, the redundancy of ZVF(n, q)is (1 −

1/q)n−1, while the approximate redundancy of Z(n, q)is

(1 −1/q)n/ln qas given in Table I. Hence, the redundancy

of the proposed VF-scheme ZVF(n, q)is roughly a factor

qln(q)/(q−1)

higher than the redundancy of Z(n, q). Note that this factor

does not depend on the code length n, but only on the alphabet

size q. For the binary case q= 2 this factor is 2 ln(2) = 1.39,

for the quaternary case q= 4 it is (4/3) ln(4) = 1.85, while

for large values of qit is roughly ln(q).

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B. Systematic Pearson Coding

An extremely simple FF scheme, called TFF(n, q), resistant

against both offset and gain mismatch, is to ﬁll the ﬁrst n−2

positions in the code sequence xwith information symbols and

to reserve the last two symbols for reference purposes: xn−1=

0and xn= 1. The resulting code sequence is in T(n, q)since

it contains at least one ‘0’ and at least one ‘1’. The redundancy

of this scheme is ﬁxed at 2 symbols, but, again, it would

be desirable to have a systematic scheme with a redundancy

decreasing in the code length, preferably approaching zero for

large values of n.

The ﬁrst VF Pearson scheme, called TVF(n, q), we propose

is similar to the VF scheme ZVF(n, q)presented in the

previous subsection. It reads as follows.

1) Take n−2information from the q-ary source and set

these as (x1, x2, . . . , xn−2).

2) If xi= 0 for at least one 1≤i≤n−2, then choose

xn−1to be a (new) information symbol, otherwise set

xn−1= 0.

3) If xi= 1 for at least one 1≤i≤n−1, then choose xn

to be a (new) information symbol, otherwise set xn= 1.

Since any code sequence obtained this way contains at least

one ‘0’ and at least one ‘1’, it is a member of T(n, q). Also, the

n−2,n−1, or ninformation symbols can easily be retrieved

from the code sequence. The redundancy of this scheme is

given in the next theorem.

Theorem 2. For a memoryless uniform q-ary source, the

redundancy of coding scheme TVF(n, q)is

2q−1

qq−1

qn−2

+1

qq−2

qn−2

.

Proof: The probability that a code sequence xhas two

redundant symbols is

(1 −2/q)n−2,(7)

which is the probability of having an information sequence of

length n−2without zeroes and ones. Further, the probability

that xhas only a redundant symbol in position n−1is

(1 −1/q)n−2−(1 −2/q)n−2,(8)

which is the probability of having an information sequence

of length n−2without zeroes but with at least one ‘1’. The

probability that xhas only a redundant symbol in position n

is

(1 −1/q)n−2−(1 −2/q)n−2(1 −1/q),(9)

where the ﬁrst multiplicative term is the probability of having

an information sequence of length n−2without ones but

with at least one ‘0’ and the second multiplicative term is the

probability that the information symbol in position n−1is

not equal to ‘1’. Hence, the redundancy is two times the term

in (7) plus the terms in (8) and (9), which gives the expression

stated in the theorem.

The redundancy of TVF(n, q)as stated in Theorem 2 is, for

large values of n, a factor

q(2q−1)

2(q−1)2ln(q)

higher than the redundancy of T(n, q)as stated in Table I.

For the binary case q= 2 this factor is 3 ln(2) = 2.08, for

the quaternary case q= 4 it is (14/9) ln(4) = 2.16, while for

large values of qit is roughly ln(q).

The second VF Pearson scheme, called PVF(n, q), we

propose is based on relaxing the enforcement of having both

at least one ‘0’ and at least one ‘1’ in all code sequences to

the enforcement that all code sequences xcontain at least one

‘0’ and have the greatest common divisor (GCD) of the xi

equal to one, i.e., GCD{x1, . . . , xn}= 1. It reads as follows.

1) Take n−2information from the q-ary source and set

these as (x1, x2, . . . , xn−2).

2) If xi= 0 for at least one 1≤i≤n−2, then choose

xn−1to be a (new) information symbol, otherwise set

xn−1= 0.

3) If GCD{x1, . . . , xn−1}= 1, then choose xnto be a

(new) information symbol, otherwise set xn= 1.

Any code sequence obtained in this way is a member of

P(n, q). Again, the n−2,n−1, or ninformation symbols

can easily be retrieved from the code sequence. For q= 2 and

q= 3, the scheme PVF(n, q)is the same as TVF(n, q), since

the condition that a sequence has a GCD of 1 is then equivalent

to the condition that a sequence contains a ‘1’. Therefore, the

redundancy is as stated in Theorem 2 in these cases. However,

this is not the case if q≥4, for which we give the redundancy

of PVF(n, q)in the next theorem. First, we present a lemma,

of which the proof is summarized due to lack of space.

Lemma 1. For any ﬁxed q≥4, among the qnq-ary sequences

yof length n, there are

1) qn−(q−1)n+O(⌈q/2⌉n)sequences with GCD(y) = 1

containing at least one ‘0’,

2) O(⌈q/2⌉n)sequences with GCD(y)̸= 1 containing at

least one ‘0’,

3) (q−1)n+O(⌊(q−1)/2⌋n)sequences with GCD(y) = 1

containing no symbol ‘0’,

4) O(⌊(q−1)/2⌋n)sequences with GCD(y)̸= 1 contain-

ing no symbol ‘0’.

Proof: The ﬁrst result was proved in [10]. Combining this

with the fact that the number of q-ary sequence of length n

containing at least one ‘0’ is qn−(q−1)ngives the second

result.

Using a well-known counting argument from, e.g., Section

16.5 in [1], it follows that the number of sequences of length

nwith symbols from {1,2, . . . , q −1}and GCD equal to 1 is

q−1

d=1

µ(d)⌊(q−1)/d⌋n= (q−1)n+O(⌊(q−1)/2⌋n),

where µ(d)is the M¨

obius function already mentioned at the

end of Subsection II-C. This proves the third result, which

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combined with the fact that the number of q-ary sequence of

length ncontaining no symbol ‘0’ is (q−1)nalso gives the

fourth result.

Theorem 3. For a memoryless uniform q-ary source, with

ﬁxed q≥4, the redundancy of coding scheme PVF(n, q)is

q−1

qn−2

+O⌈q/2⌉

qn−2.

Proof: The probability that a code sequence xhas two

redundant symbols is

O⌊(q−1)/2⌋

qn−2,(10)

which is the probability of having an information sequence of

length n−2without zeroes and with a GCD unequal to 1,

as follows from result 4) in Lemma 1. Further, the probability

that xhas only a redundant symbol in position n−1is

q−1

qn−2

+O⌊(q−1)/2⌋

qn−2,(11)

which is the probability of having an information sequence of

length n−2without zeroes but with a GCD equal to 1, as

follows from result 3) in Lemma 1. The probability that xhas

only a redundant symbol in position nis

O⌈q/2⌉

qn−2,(12)

as follows from result 2) in Lemma 1. Hence, the redundancy

is two times the term in (10) plus the terms in (11) and (12),

which gives the expression stated in the theorem.

The redundancy of PVF(n, q)as stated in Theorem 3 is, for

ﬁxed q≥4and large values of n, a factor

q

q−12

ln(q)

higher than the redundancy of P(n, q)as stated in Table I. For

the quaternary case q= 4 this factor is (16/9) ln(4) = 2.46,

while for large values of qit is roughly ln(q). Also, note that,

again for ﬁxed q≥4and large values of n, the redundancy

of PVF(n, q)is a factor q/(q−1) higher than the redundancy

of ZVF(n, q).

IV. CON CL US IO NS

We have presented simple systematic q-ary coding schemes

which are resistant against offset as well as gain mismatch or

against offset mismatch only. Both coding for ﬁxed and coding

for variable length source sequences have been considered,

resulting in FF and VF schemes of ﬁxed code block length

n, respectively. We analyzed the redundancy of the proposed

schemes for memoryless uniform sources. The major ﬁndings

are summarized in Table III.

The redundancy of the Pearson schemes TVF(n, q)and

PVF(n, q), resistant against offset as well as gain mismatch,

approaches zero for large n, as desired. The redundancy for

TABLE III

APP ROXI MATE RE DU NDA NCY O F TH E COD ES T(n, q),P(n, q ),AND

Z(n, q)AN D TH E REL ATED F F AND V F SC HEM ES ,FOR L AR GE nAN D

FIX ED q≥4.

Redundancy Red. FF Red. VF

T(n, q) 2 (q−1

q)n/ln(q) 2 2q−1

q(q−1

q)n−2

P(n, q)(q−1

q)n/ln(q)(q−1

q)n−2

Z(n, q)(q−1

q)n/ln(q) 1 (q−1

q)n−1

both schemes is equal if q= 2,3and the redundancy of

the former scheme exceeds the redundancy of the the latter

scheme by a factor of (2q−1)/q if q≥4. Furthermore,

the redundancy of the Pearson scheme PVF(n, q)exceeds the

redundancy of the ZVF(n, q)scheme, which offers immunity

to offset mismatch only, by a factor of (2q−1)/(q−1) if

q= 2,3and by a factor of only q/(q−1) if q≥4. The

schemes TFF(n, q)and ZFF(n, q)offer extreme simplicity,

using ﬁxed training symbols in ﬁxed positions, at the price

of a redundancy which does not decrease with increasing n.

Finally, the redundancy of the presented TVF(n, q ),

PVF(n, q), and ZVF(n, q)schemes is a bit higher than the

redundancy of their T(n, q),P(n, q), and Z(n, q )associates.

However, note that the low redundancies of these codes as

such are only achieved under the assumption that all their

codewords are used equally likely, which is hard to realize for

memoryless uniform and other practical sources. In contrast,

our VF schemes come with natural simple coding mechanisms.

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