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(1, 2) Optimal codes over GF(5)Bal Kishan Dass a , Rosanna Iembo b & Sapna Jain ca Department of Mathematics , University of Delhi , Delhi , 110 007 , India E-mail:b Department of Mathematics Faculty of Engineering , University of Calabria , Rende(Cosenza) , 87036 , Italy E-mail:c Department of Mathematics , Miranda House (University of Delhi) , Delhi , 110 007 ,IndiaPublished online: 31 May 2013.

To cite this article: Bal Kishan Dass , Rosanna Iembo & Sapna Jain (2006) (1, 2) Optimal codes over GF(5), Journal ofInterdisciplinary Mathematics, 9:2, 319-326, DOI: 10.1080/09720502.2006.10700446

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(1, 2) optimal codes over GF(5) ∗

Bal Kishan Dass†

Department of MathematicsUniversity of DelhiDelhi 110 007India

Rosanna Iembo‡

Department of MathematicsFaculty of EngineeringUniversity of CalabriaRende (Cosenza) 87036Italy

Sapna Jain

Department of MathematicsMiranda House(University of Delhi)Delhi 110 007India

Abstract

In this paper, we examine the possibility of the existence of codes over GF(5) whichare optimal in a specific sense viz. which are capable of correcting single errors in the firstsub-block of length n1 and bursts of length 2(fix) in the second sub-block of length n2 ,n = n1 + n2 , and no others.

Keywords : Linear codes, optimal codes, burst, sub-blocks.

∗This work was completed during first author’s visit to Dipartimento di Matematica,Universita della Calabria, Italy during June-July, 2003 as visiting Professor.

†E-mail: dassbk@rediffmail.com‡E-mails: rosannaiembo@libero.it, r.iembo@unical.it

——————————–Journal of Interdisciplinary MathematicsVol. 9 (2006), No. 2, pp. 319–326c© Taru Publications

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320 B. K. DASS, R. IEMBO AND S. JAIN

1. Introduction

The question of the existence of perfect codes has been a matter ofmuch research activity for many years. The investigations have been donewith a view to have perfect codes for random error correction. The onlyknown Hamming metric perfect codes in addition to the well-known classof single error correcting Hamming codes (1950) and the binary repe-tition codes are Golay’s (23, 12, 7) binary code and his (11, 6, 5) ternarycode (1949). After a great deal of efforts, it has finally been proved byTietavainen (1973), and van Lint (1975) that there are no perfect Hammingmetric codes over prime-power alphabets other than the above mentionedcodes.

There is another well-known type of error in the literature called“burst error”. From applications point of view, it has been found that inmany channels burst error correcting codes are more economical to usethan random error correcting codes. Dass and Tyagi (1980) studied a classof codes by considering the code of length n subdivided into two sub-blocks of lengths n1 and n2 , n1 + n2 = n which could correct bursts oflength b1 in the first sub-block and bursts of length b2 in the second sub-block. In another paper, Dass and Tyagi (1982) considered specific valuesof b1 and b2 viz. b1 = 1 and b2 = 2 and derived a class of codes in thebinary case in which such codes turn out to be such in which number oferror vectors to be corrected equals the number of cosets in the standardarray — a well-known property of perfect codes. Such codes were termedas binary (1, 2) -optimal codes. In this paper, we explore the possibility ofexistence of such codes in the 5-ary case i.e. over GF(5) . The use of suchcodes may be economical in communication than the general codes usedto correct such errors.

In what follows, an (n, k) linear code will be taken as a subspaceof the space of all n -tuples over GF(q) . The weight of a vector and thedistance between two vectors shall be considered in the Hamming sense.

Firstly, we define a burst.

Definition 1. A burst of length b is a vector all of whose non-zero compo-nents are confined to some b consecutive components, the first and thelast of which is non-zero.

There is yet another definition of a burst due to Chien and Tang (1961)which is as follows:

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OPTIMAL CODES OVER GF(5) 321

Definition 2. A CT burst of length b is a vector all of whose non-zero com-ponents are confined to some b consecutive components, the first of whichis non-zero.

Dass (1980) made a modification in the above definition by noticingthat normally errors do not occur near the end of a vector particularlywhen burst length is large.

The definition due to Dass runs as follows:

Definition 3. A burst of length b(fix) is an n -tuple whose only non-zerocomponents are confined to some b consecutive components, the firstof which is non-zero and the number of its starting positions is the firstn− b + 1 components.

2. (1, 2) optimal codes

Dass and Tyagi (1980) gave a lower bound on the necessary numberof parity-check digits required for an (n = n1 + n2, k) linear code overGF(q) that corrects bursts of length b1(fix) in the first sub-block of lengthn1 and bursts of length b2(fix) in the second sub-block of length n2 .

Whenever one obtains a bound, it is desirable to know as to for whichvalues of the parameters the bound is realized. It was found by Dass andTyagi (1982) that in the binary case for b1 = 1 and b2 = 2, not only thebound meets with equality but to the corresponding solutions, a class ofcodes called “(1, 2) binary optimal codes” exists. In the following, weshow that over GF(5) also, by taking b1 = 1 and b2 = 2, the boundresults as a source for 5-ary (1, 2) optimal codes. These codes are optimalin the sense that the number of error vectors to be corrected in such codesequals the total number of cosets viz. 5n−k .

The bound proved by Dass and Tyagi (1980) is as follows:

Theorem. The number of parity-check digits in an (n, k) linear code correctingall bursts of length b1(fix) in the first sub-block of length n1 and all bursts oflength b2(fix) in the second sub-block of length n2 (n1 + n2 = n) is at least

logq[1 + (n1 − b1 + 1)(q− 1)qb1−1 + (n2 − b2 + 1)(q− 1)qb2−1] . (1)

Equivalently, the bound in (1) may be put as follows:

qn−k ≥1 +(n1 − b1+1)(q− 1)qb1−1 +(n2 − b2+1)(q− 1)qb2−1. (1A)

To look into the values of various parameters for which the bound is tight,we must consider inequality in (1A) with equality, viz.

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322 B. K. DASS, R. IEMBO AND S. JAIN

qn−k =1 +(n1 − b1+1)(q− 1)qb1−1 + (n2 − b2+1)(q− 1)qb2−1. (2)

Let us take q = 5 and b1 = 1, b2 = 2 in (2). We get

5n1+n2−k = 4n1 + 20n2 − 19 . (3)

We now examine the values of n1, n2 and k satisfying (3). For this,we shall assign values to n1 as 1, 2, . . . , successively and find out thecorresponding values of n2 and k .

(i) Let n1 = 1.

Then (3) reduces to

5n2−k = 4n2 − 3 . (4)

The various values of n2 and k satisfying (4) with n2 ≥ 2 (because b2 = 2)are

(n2, k)={(2, 1), (7, 5), (32, 29), (157, 153), (782, 777) and so on}.

These values show the possibility of the existence of (1 + 2, 1) , (1 + 7, 5) ,(1 + 32, 29) , (1 + 157, 153) , (1 + 782, 777), . . . (1, 2) optimal codes overGF(5) .

In the following, we show that corresponding to the first set of values,such a code exists i.e. (1 + 2, 1) 5-ary (1, 2) optimal code exists.

Example. Consider the following parity-check matrix for (1 + 2, 1) 5-arycode:

H =

[3 0 40 4 0

].

The code which is the null space of H can correct all bursts of length 1in the first sub-block of length 1 and all bursts of length 2(fix) in thesecond block of length 2 and no others. We list below all the error vectorsand their corresponding syndromes which can be seen to be distinctaltogether and exhaustive.

Error vectors SyndromesFirst sub-block

1 00 302 00 103 00 404 00 20

(Contd.)

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OPTIMAL CODES OVER GF(5) 323

Second sub-block0 10 040 11 440 12 340 13 240 14 140 20 030 21 430 22 330 23 230 24 130 30 020 31 420 32 320 33 220 34 120 40 010 41 410 42 310 43 210 44 11

(ii) Let n1 = 2.Then (3) reduces to

5n2−k+2 = 20n2 − 11 .

It is clear that R.H.S. of above equation is never a power of 5. Therefore,this equation has no integer solutions for n2 and k and hence such a (1, 2)optimal code cannot exist in this case.

(iii) Let n1 = 3.Then (3) reduces to

5n2−k+3 = 20n2 − 7 .

Once again, it is evident that this equation has no integer solutions for n2and k and thus such a (1, 2) optimal code cannot exist in this case as well.

(iv) Let n1 = 4.Then (3) reduces to

5n2−k+4 = 20n2 − 3 .

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324 B. K. DASS, R. IEMBO AND S. JAIN

This equation clearly has no integer solutions for n2 and k and thus no(1, 2) optimal code exists in this case.

(v) Let n1 = 5.Then equation (3) becomes

5n2−k+5 = 20n2 + 1 .

Once again the case of no integer solution and hence of no (1, 2) optimalcode in this case.

(vi) Let n1 = 6.Then (3) reduces to

5n2−k+5 = 4n2 + 1 . (5)

The following are the values of n2 and k satisfying (5) with n2 ≥ 2:

(n2, k) = {(31, 34), (156, 158), (781, 782), . . . and so on} .

This shows the possibility of the existence of (6 + 31, 34) , (6 + 156, 158) ,(6 + 781, 782) , . . . 5-ary (1, 2) optimal codes.

(vii) Let n1 = 7.Then equation (3) reduces to

5n2−k+7 = 20n2 + 9 .

This equation has no integer solutions for n2 and k and thus no (1, 2)optimal code exists in this case.

(viii) Let n1 = 8.Then equation (3) gives

5n2−k+8 = 20n2 + 13 ,

which is again a case of no integer solution and no (1, 2) optimal code.

(ix) Let n1 = 9.Then equation (3) gives

5n2−k+9 = 20n2 + 17 ,

a case of no integer solution and no (1, 2) optimal code.

(x) Let n1 = 10 .The equation (3) results

5n2−k+10 = 20n2 + 21 ,

again a case of no integer solution and no optimal code.

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OPTIMAL CODES OVER GF(5) 325

(xi) Let n1 = 11 .Then equation (3) gives

5n2−k+10 = 4n2 + 5 . (6)

The values of n2 and k satisfying (6) with n2 ≥ 2 are:

(n2, k) = {(5, 13), (30, 37), (155, 161), (780, 785), . . . and so on},

which once again shows the possibility of the existence of (11 + 5, 13) ,(11 + 30, 37) , (11 + 155, 161) , (11 + 780, 785), . . . 5-ary (1, 2) optimalcodes.

3. Conclusions and remarks

It is known that use of optimal codes economize in the number ofparity-check digits in any code reducing thereby the redundancy andimproves upon rate of transmission making the channels more efficient.Therefore, to look for optimal codes is useful from applications point ofview in communication.

In this paper, we have investigated solutions of equation (3) forn1 = 1, 2, . . . , 11 . It is seen that equation (3) has solutions for n1 = 1, 6and 11. It is clear that equation (3) has integer solutions for n1 = 16, 21, . . .and no integer solutions for n1 = 12, 13, 14, 15, 17, 18, 19, 20, . . . .

We have been able to obtain code corresponding to one of thesolutions only. This justifies that such a (1, 2) optimal 5-ary code exists.However, in view of the existence of other solutions of equation (3), theexistence of corresponding codes is an open problem.

References

[1] R. T. Chien and D. T. Tang (1965), On definition of a burst, IBM J. Res.& Develop., Vol. 9 (4), pp. 292–293.

[2] B. K. Dass (1980), On a burst error correcting code, J. Infor. & Optimz.Sciences, Vol. 1, pp. 291–295.

[3] B. K. Dass and V. K. Tyagi (1980), Bounds on blockwise burst errorcorrecting linear codes, Information Science, Vol. 20, pp. 157–164.

[4] B. K. Dass and V. K. Tyagi (1982), A new type of (1, 2) -optimal codesover GF(2) , Indian J. Pure & Appl. Math., Vol. 13, pp. 750–756.

[5] M. J. E. Golay (1949), Notes on digital coding, Proc. IRE, Vol. 37,p. 657.

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[6] R. W. Hamming (1950), Error detecting and error correcting codes,Bell Syst. Tech. J., Vol. 29, pp. 147–160.

[7] A. Tietavainen (1973), On the non-existence of perfect codes overfinite fields, SIAM J. Appl. Math., Vol. 24 (1), pp. 88–96.

[8] J. H. van Lint (1975), A survey of perfect codes, Rocky Mountain J.Math., Vol. 5 (2), pp. 199–226.

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