High-density-burst error detection

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High-density-burst error detectionBiagio Buccimazza a , Bal Kishan Dass b & Sapna Jain ca Dipartimento di Ingegneria Elettrica , Universita de L’Aquila , Monteluco di Roio,L’Aquila , I-67040 , Italyb Department of Mathematics , University of Delhi , Delhi , 110 007 , Indiac Department of Mathematics , University of Delhi , Miranda House, Delhi , 110 007 ,IndiaPublished online: 03 Jun 2013.

To cite this article: Biagio Buccimazza , Bal Kishan Dass & Sapna Jain (2004) High-density-burst error detection, Journalof Discrete Mathematical Sciences and Cryptography, 7:1, 5-21, DOI: 10.1080/09720529.2004.10697984

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High-density-burst error detection∗

Biagio Buccimazza

Dipartimento di Ingegneria Elettrica

Universita de L’Aquila

I-67040 Monteluco di Roio

L’Aquila

Italy

Bal Kishan Dass

Department of Mathematics

University of Delhi

Delhi 110 007

India

Sapna Jain

Department of Mathematics

Miranda House

University of Delhi

Delhi 110 007

India

Abstract

In this paper, we study cyclic codes detecting a subclass of open-loop bursts namelyHigh-density open-loop bursts. A subclass of CT open-loop bursts called CT high-densityopen-loop bursts is also studied. A comparative study of the results obtained in this paperwith the earlier known results and among themselves has also been made.

∗This work was finally finalized during second author’s visits to University of L’Aquila(Italy) under G.N.S.A.G.A. of I.N.D.A.M. program in May-June 2000 and September-October 2002.

—————————————————–Journal of Discrete Mathematical Sciences & CryptographyVol. 7 (2004), No. 1, pp. 5–21c© Taru Publications

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6 B. BUCCIMAZZA, B. K. DASS AND S. JAIN

1. Introduction

In several communication systems, errors occur predominantly in theform of bursts. Codes developed to detect and correct such errors havebeen studied extensively by many authors. Most of the studies in thisdirection have been made with respect to the following definition of aburst (Peterson and Weldon (1972)) :

Definition 1. A burst of length b is a vector all of whose non zerocomponents are confined to some b consecutive components, the first andthe last of which is non zero.

Such bursts are sometimes referred to as ‘open-loop bursts’. It is wellknown that cyclic codes are well suited for error detection. It is also knownthat in an (n, k) cyclic code a burst (according to Definition 1) of length(n− k) or less is always detectable. However, there are several bursts oflength greater than n− k which go undetected. Two known results in thisdirection are stated below :

Theorem A. No code vector of an (n, k) cyclic code is a burst of length n− k orless. Therefore, every (n, k) cyclic code can detect any burst of length n− k orless.

Theorem B. The fraction of bursts of length b > n− k that can be undetectedby an (n, k) cyclic code is

q−(n−k−1)/(q− 1) if b = n− k + 1

q−(n−k) if b > n− k + 1 .

For the proofs of Theorem A and B, one may refer to Peterson andWeldon (1972, pp. 229-230). Some implications of Theorem B have beenpointed out by Brown and Peterson (1961) .

An application of these ideas to a practical error detection problemhas been described by Fontaine and Gallager (1960).

There is another definition of a burst due to Chien and Tang (1965)which runs as follows :

Definition 2. A burst of length b is a vector all of whose non zerocomponents are confined to some b consecutive components, the first ofwhich is non zero.

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BURST ERROR DETECTION 7

Such bursts are termed as CT bursts.

Dass and Jain (2000) proved the following result for CT open-loopbursts.

Theorem C. The fraction of CT open-loop bursts of length b > n− k that canbe undetected by an (n, k) cyclic code over GF(q) is q−(n−k) .

In certain systems lightening or other disturbances which introduceburst errors usually operate in such a way that over a given length, somedigits are received correctly, although others are corrupted. Also, when theburst length is large, the actual number of errors inside the burst lengthis very less. Such situations demand the development of codes whichdetect/correct those errors that are bursts of length b or less with weightw or more (2 ≤ w ≤ b) . Such bursts will be termed as high-density open-loop bursts. The development of such codes economize in the parity checkdigits required, suitably reduce the redundancy of the code i.e. suitablyincrease the efficiency of transmission. In the second and third section ofthis paper we obtain results similar to Theorems A and B for high-densityopen-loop bursts and CT high-density open-loop bursts, respectively. Insection 4, we make a comparative study of the results derived in this paperwith the earlier known results as well as among themselves.

In what follows, an (n, k) cyclic code over GF(q) is taken as an idealin the algebra of polynomials modulo the polynomial Xn − 1 .

2. High-density open-loop burst error detection

In this section, we obtain the results of Theorems A and B for high-density open-loop bursts.

Theorem 1. An (n, k) cyclic code can detect high-density open-loop bursts oflength n− k or less having weights w or more (2 ≤ w ≤ b) .

Proof. There is no deviation in the final conclusion from that of Theorem Abecause the proof is based on the length of the burst giving rise to apolynomial which remains of the same degree even when the weightconsideration over the burst is taken. Hence the proof is omitted. ¤

Theorem 2. The fraction of high-density open-loop bursts of length b(> n− k)having weight w or more (2 ≤ w ≤ b) that goes undetected in any (n, k) cycliccode over GF(q) is

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1b−2

∑i=w−2

(b− 2

i

)(q− 1)i+1

when b = n− k + 1

qb−(n−k)−2

b−2

∑i=w−2

(b− 2

i

)(q− 1)i

when b > n− k + 1 .

Proof. Consider r(X) to be a high-density open-loop burst of lengthb(> n− k) having weight ≥ w(w ≤ b) .

Let g(X) be the generator polynomial of the code.

Obviously, deg g(X) = n− k .

Then r(X) = Xi r1(X) for some non negative integer i and for somepolynomial r1(X) where deg r1(X) = b− 1 and there are at least w− 2non zero coefficients in r1(X) apart from the coefficient of highest degreeterm and the constant nonzero term.

We have

# of high-density open-loop bursts of length b having weight w ormore (2 ≤ w ≤ b)

= # of polynomials of type r(X)

= # of polynomials of type r1(X)

= (q− 1)2b−2

∑i=w−2

(b− 2

i

)(q− 1)i

Now r(X) will go undetected if g(X) | r(X)

⇒ g(X) must divide Xi r1(X)

⇒ g(X) must divide r1(X) , since g(X) and Xi are coprime.

Let r1(X) = g(X) Q(X) for some polynomial Q(X) .

Since deg r1(X) = b− 1

and deg g(X) = n− k

∴ deg Q(X) = (b− 1)− (n− k) .

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Two cases arise :

Case 1. When b = n− k + 1 .

Then deg Q(X) = 0

⇒ Q(X) is a non zero constant

⇒ No of possibilities for Q(X) equals q− 1 .

∴ The ratio of high-density open-loop bursts which go undetected tothe total number of high-density open-loop bursts is

=(q− 1)

(q− 1)2b−2

∑i=w−2

(b− 2

i

)(q− 1)i

=1

(q− 1)b−2

∑i=w−2

(b− 2

i

)(q− 1)i

=1

b−2

∑i=w−2

(b− 2

i

)(q− 1)i+1

.

Case 2. When b > n− k + 1 .

Then # of polynomials of type Q(X)

= # of polynomials of degree (b− 1)− (n− k)

= (q− 1)2 q(b−1)−(n−k)−1 (∵ first and last coefficient of Q(X) mustbe non zero)

∴ Ratio in this case

=(q− 1)2 q(b−1)−(n−k)−1

(q− 1)2b−2

∑i=w−2

(b− 2

i

)(q− 1)i

=qb−(n−k)−2

b−2

∑i=w−2

(b− 2

i

)(q− 1)i

which proves the theorem. ¤

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Special Case. For b = 2 , w = 2 , the above theorem reduces to Theorem Afor b = 2 .

4. CT high-density open-loop burst error detection

In this section we extend the studies made in Section 2 for CT high-density open-loop bursts.

We begin by stating the following result without proof.

Theorem 3. Every (n, k) cyclic code detects any CT high-density open-loopburst of length b(b ≤ n− k) having weight ≥ w(w ≤ b) .

Now we prove the following result.

Theorem 4. The fraction of CT high-density open-loop bursts of length b >

n − k having weight ≥ w(w ≤ b) that can go undetected in an (n, k) cycliccode is

=q(b−1)−(n−k)

b−1

∑i=w−1

(b− 1

i

)(q− 1)i

.

Proof. Consider r(X) to be a CT high-density open-loop burst of lengthb(> n− k) having weight ≥ w(w ≤ b) .

Let g(X) be the generator of the code, then deg g(X) = n− k .

As earlier, r(X) = Xi ri(X) for some non negative integer i and for somepolynomial r1(X) where deg r1(X) ≤ b− 1 and there are at least w− 1non zero coefficients in r1(X) apart from constant (non zero) term.

(Note. The highest degree term in r1(X) may be zero by the definition ofCT burst.)

Now # CT high-density open-loop bursts of length b having weight≥ w(w ≤ b)

= # of polynomials of type r(X)

= # of polynomials of type r1(X)

= (q− 1)b−1

∑i=w−1

(b− 1

i

)(q− 1)i .

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Now r(X) will go undetected if g(X) | r(X)

⇒ g(X) must divide Xir1(X)

⇒ g(X) must divide r1(X) , since g(X) and Xi are coprime.

Let r1(X) = g(X) Q(X) for some polynomial Q(X) .

Since deg r1(X) ≤ b− 1 and deg g(X) = n− k

∴ deg Q(X) ≤ (b− 1)− (n− k) .

Two cases arise :

Case 1. When b = n− k + 1 .

Then deg Q(X) ≤ 0

⇒ deg Q(X) = 0 , as degree of a polynomial can not be negative

⇒ Q(X) is a non zero constant

⇒ # of possibilities for Q(X) = q− 1

∴ The ratio of CT high-density open-loop bursts that go undetected tothe total number of CT high-density open-loop bursts is

=(q− 1)

(q− 1)b−1

∑i=w−1

(b− 1

i

)(q− 1)i

=1

b−1

∑i=w−1

(b− 1

i

)(q− 1)i

.

Case 2. When b > n− k + 1 , i.e., b− 1 > n− k .

Then # of polynomials of type Q(X)

= # of polynomials of degree ≤ (b− 1)− (n− k)

= (q− 1) q(b−1)−(n−k)

∴ Ratio in this case

=(q− 1) q(b−1)−(n−k)

(q− 1)b−1

∑i=w−1

(b− 1

i

)(q− 1)i

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=q(b−1)−(n−k)

b−1

∑i=w−1

(b− 1

i

)(q− 1)i

combing the two cases, we get that

Ratio =q(b−1)−(n−k)

b−1

∑i=w−1

(b− 1

i

)(q− 1)i

.

Hence the theorem. ¤

Special Case. For b = 1 , w = 1 , the above theorem reduces to Theorem Cfor b = 1 .

5. Comparative study

In this section we first present the comparisons of the result obtainedin Section 2 and Section 3 viz. Theorem 2 and Theorem 3 and then wepresent the comparison of the result obtained in Section 2 viz. Theorem 2with the earlier known result i.e. Theorem B. The comparisons have beenpresented in the form of tables by taking specific values of b ; b and w inthe binary as well as in the non binary cases.

Table 1High-density open-loop and CT high-density open-loop bursts

Fraction for high-density open-loop burst

=

1b−2

∑i=w−2

(b− 2

i

)(q− 1)i+1

when b = n− k + 1

qb−(n−k)−2

b−2

∑i=w−2

(b− 2

i

)(q− 1)i

when b > n− k + 1

Fraction for CT high density open loop bursts

=q(b−1)−(n−k)

b−1

∑i=w−1

(b− 1

i

)(q− 1)i

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High-density open-loop CT high-density open-loop(Theorem 2) (Theorem 4)

q = 2, b = 2 ⇒ w = 2

n− k + 1 = 2 1 1

n− k + 1 < 2 2−(n−k) 21−(n−k)/1

for k = 1, 2, 3 the above reduces to

n = 2 1 1 for k = 1

n = 3 1 1}

for k = 2n < 3 22−n 23−n

n = 4 1 1}

for k = 3n < 4 23−n 24−n

q = 2, b = 3 ⇒ w = 2, 3

w = 2

n− k + 1 = 3 1/2 1/3

n− k + 1 < 3 2−(n−k) 22−(n−k)/3


n = 3 1/2 1/3 for k = 1

n = 4 1/2 1/3}

for k = 2n < 4 22−n 24−n/3

n = 5 1/2 1/3}

for k = 3n < 5 23−n 25−n/3

w = 3

n− k + 1 = 3 1 1

n− k + 1 < 3 21−(n−k) 22−(n−k)


n = 3 1 1 for k = 1

n = 4 1 1}

for k = 2n < 4 2−(n−3) 2−(n−4)

n = 5 1 1}

for k = 3n < 5 2−(n−4) 2−(n−5)

(Table 1 Contd.)

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q = 2, b = 4 ⇒ w = 2, 3, 4

w = 2

n− k + 1 = 4 1/4 1/7

n− k + 1 < 4 2−(n−k) 23−(n−k)/7


n = 4 1/4 1/7 for k = 1

n = 5 1/4 1/7}

for k = 2n < 5 22−n 25−n/7

n = 6 1/4 1/7}

for k = 3n < 6 23−n 26−n/7

w = 3

n− k + 1 = 4 1/3 1/4

n− k + 1 < 4 22−(n−k)/3 23−(n−k)/4 = 21−(n−k)


n = 4 1/3 1/4 for k = 1

n = 5 1/3 1/4}

for k = 2n < 5 24−n/3 23−n

n = 6 1/3 1/4}

for k = 3n < 6 25−n/3 24−n

w = 4

n− k + 1 = 4 1 1

n− k + 1 < 4 22−(n−k) 23−(n−k)


n = 4 1 1 for k = 1

n = 5 1 1}

for k = 2n < 5 24−n 25−n

n = 6 1 1}

for k = 3n < 6 25−n 26−n

(Table 1 Contd.)

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q = 2, b = 5 ⇒ w = 2, 3, 4, 5

w = 2

n− k + 1 = 5 1/8 1/15

n− k + 1 < 5 2−(n−k) 24−(n−k)/15


n = 5 1/8 1/15 for k = 1

n = 6 1/8 1/15}

for k = 2n < 6 22−n 26−n/15

n = 7 1/8 1/15}

for k = 3n < 7 23−n 27−n/15

w = 3

n− k + 1 = 5 1/7 1/11

n− k + 1 < 5 23−(n−k)/7 24−(n−k)/11


n = 5 1/7 1/11 for k = 1

n = 6 1/7 1/11}

for k = 2n < 6 25−n/7 26−n/11

n = 7 1/7 1/11}

for k = 3n < 7 26−n/7 27−n/11

w = 4

n− k + 1 = 5 1/4 1/5

n− k + 1 < 5 21−(n−k) 24−(n−k)/5


n = 5 1/4 1/5 for k = 1

n = 6 1/4 1/5}

for k = 2n < 6 23−n 26−n/5

n = 7 1/4 1/5}

for k = 3n < 7 24−n 27−n/5

(Table 1 Contd.)

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w = 5

n− k + 1 = 5 1 1

n− k + 1 < 5 23−(n−k) 24−(n−k)


n = 5 1 1 for k = 1

n = 6 1 1}

for k = 2n < 6 25−n 26−n

n = 7 1 1}

for k = 3n < 7 26−n 27−n

q = 3, b = 3 ⇒ w = 2, 3

w = 2

n− k + 1 = 3 1/6 1/8

n− k + 1 < 3 3−(n−k) 32−(n−k)/8


n = 3 1/6 1/8 for k = 1

n = 4 1/6 1/8}

for k = 2n < 4 32−n 34−n/8

n = 5 1/6 1/8}

for k = 3n < 5 33−n 35−n/8

w = 3

n− k + 1 = 3 1/4 1/4

n− k + 1 < 3 31−(n−k)/2 32−(n−k)/4


n = 3 1/4 1/4 for k = 1

n = 4 1/4 1/4}

for k = 2n < 4 33−n/2 34−n/4

n = 5 1/4 1/4}

for k = 3n < 5 34−n/2 35−n/4

(Table 1 Contd.)

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q = 3, b = 4 ⇒ w = 2, 3, 4

w = 2

n− k + 1 = 4 1/18 1/26

n− k + 1 < 4 32−(n−k)/9 33−(n−k)/26


n = 4 1/18 1/26 for k = 1

n = 5 1/18 1/26}

for k = 2n < 5 34−n/9 35−n/26

n = 6 1/18 1/26}

for k = 3n < 6 35−n/9 36−n/26

w = 3

n− k + 1 1/16 1/20

n− k + 1 < 4 32−(n−k)/8 33−(n−k)/20


n = 4 1/16 1/20 for k = 1

n = 5 1/16 1/20}

for k = 2n < 5 34−n/8 35−n/20

n = 6 1/16 1/20}

for k = 3n < 6 35−n/8 36−n/20

w = 4

n− k + 1 = 4 1/8 1/8

n− k + 1 < 4 32−(n−k)/4 33−(n−k)/8


n = 4 1/8 1/8 for k = 1

n = 5 1/8 1/8}

for k = 2n < 5 34−n/4 35−n/8

n = 6 1/8 1/8}

for k = 3n < 6 35−n/4 36−n/8

(Table 1 Contd.)

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q = 3, b = 5 ⇒ w = 2, 3, 4, 5

w = 2

n− k + 1 = 5 1/54 1/80

n− k + 1 < 5 33−(n−k)/27 34−(n−k)/80


n = 5 1/54 1/80 for k = 1

n = 6 1/54 1/80}

for k = 2n < 6 35−n/27 36−n/80

n = 7 1/54 1/80}

for k = 3n < 7 36−n/27 37−n/80

w = 3

n− k + 1 = 5 1/52 1/72

n− k + 1 < 5 33−(n−k)/26 34−(n−k)/72


n = 5 1/52 1/72 for k = 1

n = 6 1/52 1/72}

for k = 2n < 6 35−n/26 36−n/72

n = 7 1/52 1/72}

for k = 3n < 7 36−n/26 37−n/72

w = 4

n− k + 1 = 5 1/40 1/48

n− k + 1 < 5 33−(n−k)/20 34−(n−k)/48


n = 5 1/40 1/48 for k = 1

n = 6 1/40 1/48}

for k = 2n < 6 35−n/20 36−n/48

n = 7 1/40 1/48}

for k = 3n < 7 36−n/20 37−n/48

(Table 1 Contd.)

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w = 5

n− k + 1 = 5 1/16 1/16

n− k + 1 < 5 33−(n−k)/8 34−(n−k)/16


n = 5 1/16 1/16 for k = 1

n = 6 1/16 1/16}

for k = 2n < 6 35−n/8 36−n/16

n = 7 1/16 1/16}

for k = 3n < 7 36−n/8 37−n/16

Table 2

Open-loop bursts and high density open loop bursts

Fraction for open-loop bursts

q−(n−k−1)/(q− 1) if b = n− k + 1

q−(n−k) if b > n− k + 1

Fraction for high-density open-loop bursts

=

1b−2

∑i=w−2

(b− 2

i

)(q− 1)i+1

when b = n− k + 1

qb−(n−k)−2

b−2

∑i=w−2

(b− 2

i

)(q− 1)i

when b > n− k + 1

Open-loop High-density open-loop(Theorem B) (Theorem 2)

q = 2, b = 3

n− k + 1 = 3 1/2 1/2 for w = 2

1 for w = 3

n− k + 1 < 3 2−(n−k) 2−(n−k) for w = 2

21−(n−k) for w = 3

(Table 1 Contd.)

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q = 2, b = 4

n− k + 1 = 4 1/4 1/4 for w = 2

1/3 for w = 3

1 for w = 4

n− k + 1 < 4 2−(n−k) 2−(n−k) for w = 2

22−(n−k)/3 for w = 3

22−(n−k) for w = 4

q = 2, b = 5

n− k + 1 = 5 1/8 1/8 for w = 2

1/7 for w = 3

1/4 for w = 4

1 for w = 5

n− k + 1 < 5 2−(n−k) 2−(n−k) for w = 2

23−(n−k)/7 for w = 3

21−(n−k) for w = 4

23−(n−k) for w = 5

q = 3, b = 3

n− k + 1 = 3 1/6 1/6 for w = 2

1/4 for w = 3

n− k + 1 < 3 3−(n−k) 3−(n−k) for w = 2

31−(n−k)/2 for w = 3

q = 3, b = 4

n− k + 1 = 4 1/18 1/18 for w = 2

1/16 for w = 3

1/8 for w = 4

n− k + 1 < 4 3−(n−k) 32−(n−k)/9 for w = 2

32−(n−k)/8 for w = 3

32−(n−k)/4 for w = 4

(Table 2 Contd.)

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q = 3, b = 5

n− k + 1 = 5 1/54 1/54 for w = 2

1/52 for w = 3

1/40 for w = 4

1/16 for w = 5

n− k + 1 < 5 3−(n−k) 33−(n−k)/27 for w = 2

33−(n−k)/26 for w = 3

33−(n−k)/20 for w = 4

33−(n−k)/8 for w = 5

Aacknowledgement. The third author wishes to thank University GrantsCommission for providing grant (vide Ref. No. F.13-3/99 (SR-I)) underMinor Research Project to carry out this research work.

References

[1] A. A. Alexander, R. M. Gryb and D. W. Nast (1960), Capabilities ofthe Telephone Network for Data Transmission, Bell Syst. Tech. J., Vol.39 (3), May.

[2] D. T. Brown and W. W. Peterson (1961), Cyclic Codes for ErrorDetection, Proc. IRE, Vol. 49, pp. 228-235.

[3] R. T. Chien and D. T. Tang (1965), On Definitions of a Burst, IBM J.Res. & Develop., July, pp. 292-293.

[4] B. K. Dass and Sapna Jain (2000), Various Burst Errors DetectingCodes, Italian J. Pure & Appl. Math., to appear.

[5] A. B. Fontaine and R. G. Gallager (1960), Error-Statistics and Codingfor Binary Transmission over Telephone Circuits, Technical Report25G-0023, M.I.T. Lincoln Laboratory, Lexington, Massachusetts,October.

[6] W. W. Peterson and E. J. Weldon, Jr. (1972), Error-Correcting Codes,The MIT Press, Massachusetts.

Received October, 2002

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High-density-burst error detection