The objective of coding theory is to protect a message going through a noisy channel. The nature of errors that cause noisy channel depends on different factors. Accordingly codes are needed to develop to deal with different types of errors. Sharma and Gaur [6] introduced a new kind of error which is termed as ‘key error’. This paper presents lower and upper bounds on the number of parity-check digits required for linear codes capable of correcting such errors. An example of such a code is also provided.Publisher's Versio

Das, Pankaj Kumar

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.5, N.1, 2015, pp. 110-117.CODES CORRECTING KEY ERRORSPANKAJ KUMAR DAS1, §Abstract. The objective of coding theory is to protect a message going through a noisychannel. The nature of errors that cause noisy channel depends on different factors.Accordingly codes are needed to develop to deal with different types of errors. Sharmaand Gaur [6] introduced a new kind of error which is termed as ‘key error’. This paperpresents lower and upper bounds on the number of parity-check digits required for linearcodes capable of correcting such errors. An example of such a code is also provided.Keywords: Parity check matrix, syndrome, standard array, solid burst error.AMS Subject Classification (2010): 94B05, 94B65.1. IntroductionWith the advancement of information technology, different types of new problems havebeen coming out. Different types of error patterns are also coming out. Error controlcoding now is not limited to distant communication only. There are communication chan-nels like the automata or electronics devices which are encountered with specific type oferrors for which the coding is required. The error patterns which they produce are alldependent on the characteristics of the device. It is important to carefully study the errorpatterns that actually occur. This allows correction of all errors that actually occur ratherthan partial correction or correcting non-errors that results in wasting the capacity of thechannel.Let us consider the keyboard of a computer; it has keys for various numbers and othersymbols. Imagine punching a number or an alphabet key on it. While word processing,one may erroneously press a key on one or two positions on either side of the right key,rather than any key on the keyboard. These likely positions will constitute the set oferrors for the number or the symbol key pressed. Such errors are already discussed bySharma and Gaur [6] with respect to S-K metric. We give here a study on such errorswith respect to Hamming metric. We call such errors as key errors and they are definedas follows:Definition 1.1. A i-key error of length b is a vector such that the ith component is non-zero and the other non-zero components are confined to some b consecutive components ineither side of the ith component.1 Department of Mathematics, Shivaji College(University of Delhi), Raja Garden, Delhi-110 027, India.e-mail: pankaj4thapril@yahoo.co.in;§ Manuscript received: July 05, 2014.TWMS Journal of Applied and Engineering Mathematics, Vol.5, No.1; c⃝ Işık University, Departmentof Mathematics, 2015; all rights reserved.110P. K. DAS: CODES CORRECTING KEY ERRORS 111It may be noted that in such error, the entry error i.e., the ith component may be thefirst position and go to upto the nth position. If the entry error is the first position, thenthe non-zero components are confined to b consecutive components on the right side ofthe first position. If the entry error is the second position, then non-zero components areconfined to one position of left side of the second position and to the b consecutive com-ponents on the right side of the second position. Continue the process such that nonzerocomponents are confined to b consecutive positions of either side of the entry error. Fi-nally if the entry error is the nth position, then non-zero components are confined to theb consecutive components on the left side of the nth component.For example in a vector of length 6 over a field of 3 elements GF (3), the key er-rors of length 2 are (0 12︸︷︷︸22︸︷︷︸entry error12︸︷︷︸2), (0 12︸︷︷︸22︸︷︷︸entry error02︸︷︷︸2), (0 12︸︷︷︸22︸︷︷︸entry error00︸︷︷︸2),(000 12︸︷︷︸22︸︷︷︸entry error), ( 12︸︷︷︸21︸︷︷︸entry error12︸︷︷︸20), ( 2︸︷︷︸12︸︷︷︸entry error12︸︷︷︸200), etc.The basic purpose of error-correcting codes is to correct errors that are occurred duringcommunication. This is done by adding parity check digits (redundancy) to the infor-mation digits. The efficiency of a error correcting code depends on the number of paritycheck digits. The lesser is the number of parity check symbols in a code, the more isthe rate of information of the code. It is not generally possible to give the exact numberof redundant/parity check digits for a given error correcting code. But, we can obtainbounds on the number of redundant/parity check digits. This was initiated by Hamming[2] who was concerned with both code constructions and bounds. After that, many otherresearchers have worked on bounds and this paper is also in that direction.Das [1] has presented lower and upper bounds on parity check for the codes detectingkey errors. Reiger type of bound [4] on codes detecting and simultaneously correctingsuch errors was also studied. The present paper obtains lower and upper bounds on paritycheck digits for a linear code capable of correcting such errors.The paper is organized as follows: Section 1 i.e., the Introduction gives brief view of theimportance of the study of the paper and basic definition. In Section 2, we obtain lowerand upper bounds on the number of parity check digits of a linear code that corrects anykey error of length b or less. This is followed by an example of such a code. Section 3 isthe conclusion.In what follows a linear code will be considered as a subspace of the space of all n-tuplesover GF (q). The distance between two vectors shall be considered in the Hamming sense.2. Correction of key errorsWe consider the linear codes that are capable of correcting any key errors of length bor less. Firstly, a lower bound on the number of parity-check digits required for such acode is obtained. The proof is based on the technique used in Theorem 4.16, Peterson andWeldon [3].Theorem 2.1. Any (n, k) linear code over GF (q) that corrects any key error of length bor less must satisfyqn−k ≥ 1 + q2b+1 − qq + 1+ (n− 2b)(q − 1)(q2b+1 − qq + 1+ 1)+q2b+1 − q3(q + 1)2+(q − 1)(b+ q)q + 1.Proof. The result will be proved by counting the number of correctable errors and settingit less than or equal to qn−k.112 TWMS J. APP. ENG. MATH. V.5, N.1, 2015The number of key errors of length b or less when the entry error of the key errors is from1st position to bth position is given by(q − 1)qb+(q − 1)2qb + (q − 1)qb−1+(q − 1)2qb+1 + (q − 1)2qb−1 + (q − 1)qb−2+(q − 1)2qb+2 + (q − 1)2qb + (q − 1)2qb−2 + (q − 1)qb−3+........................................+.........................................+(q − 1)2q2b−2 + (q − 1)2q2b−4 + (q − 1)2q2b−6 + · · ·+ (q − 1)2q2 + (q − 1)q,which is equal to(q − 1)2{qb(q2 − 1q2 − 1)+ qb−1((q2)2 − 1q2 − 1)+ qb−2((q2)3 − 1q2 − 1)+ · · ·+ q3((q2)b−2 − 1q2 − 1)+ q2((q2)b−1 − 1q2 − 1)}+ (q − 1)q(qb − 1q − 1),which on simplification givesq2b+1 − qq + 1. (1)The number of key errors of length b or less when the entry error of the key errors is from(b+ 1)th position to (n− b)th position is given by(n− 2b){(q − 1)2q2b−1 + (q − 1)2q2b−3 + (q − 1)2q2b−5 + · · ·+ (q − 1)2q + (q − 1)}= (n− 2b)(q − 1)(q2b+1 − qq + 1+ 1). (2)If the entry error of the key errors is in the last b positions, the number of key errors oflength b or less is(q − 1)2q2b−3 + (q − 1)2q2b−5 + (q − 1)2q2b−7 + · · ·+ (q − 1)2q3 + (q − 1)2q + (q − 1)+(q − 1)2q2b−5 + (q − 1)2q2b−7 + (q − 1)2q2b−9 + · · ·+ (q − 1)2q3 + (q − 1)2q + (q − 1)+.........................................+.........................................+(q − 1)2q5 + (q − 1)2q3 + (q − 1)2q + (q − 1)+(q − 1)2q3 + (q − 1)2q + (q − 1)+(q − 1)2q + (q − 1)+(q − 1),which is equivalent to(q − 1)2{q((q2)b−1 − 1q2 − 1)+ q((q2)b−2 − 1q2 − 1)+ q((q2)b−3 − 1q2 − 1)+ · · ·+ q((q2)2 − 1q2 − 1)+ q(q2 − 1q2 − 1)}+ (q − 1)b,P. K. DAS: CODES CORRECTING KEY ERRORS 113which again on simplification givesq2b+1 − q3(q + 1)2−(q − 1q + 1)q(b− 1) + (q − 1)b=q2b+1 − q3(q + 1)2+(q − 1)(b+ q)q + 1. (3)Therefore, the total number of key errors of length b or less is given byexpr.(1) + expr.(2) + expr.(3)=q2b+1 − qq + 1+ (n− 2b)(q − 1)(q2b+1 − qq + 1+ 1)+q2b+1 − q3(q + 1)2+(q − 1)(b+ q)q + 1.For correction, all these vectors must belong to different cosets. The total number of cosetsavailable is qn−k . Therefore, we must haveqn−k ≥ 1 + q2b+1 − qq + 1+ (n− 2b)(q − 1)(q2b+1 − qq + 1+ 1)+q2b+1 − q3(q + 1)2+(q − 1)(b+ q)q + 1.Now the following theorem gives an upper bound on the number of check digits requiredfor the construction of a linear code considered in Theorem 2.1. This bound assuresthe existence of a linear code that can correct all key errors of length b or less. Theproof is based on the well known technique used in Varshomov-Gilbert Sacks bound byconstructing a parity check matrix for such a code (refer Sacks [5], also Theorem 4.7Peterson and Weldon [3]). The procedure involves suitable modifications of the techniqueused in deriving Varshamov-Gilbert-Sacks bound.Theorem 2.2. There shall always exist an (n, k) linear code over GF (q) that corrects anykey error of length b or less (n > 4b+ 2) provided thatqn−k >(q2b+1 + 1q + 1){1 +q2b+1 − qq + 1+ (n− 1− 4b)(q − 1)(q2b+1 − qq + 1+ 1)+q2b+1 − q3(q + 1)2+(q − 1)(b+ q)q + 1}.Proof. The existence of such a code will be proved by constructing an (n− k)× n paritycheck matrix H for the desired code as follows:Select any non zero (n − k)-tuple as the first column h1 of the matrix H. After havingselected the first j − 1 columns h1, h2, . . . , hj−1 appropriately, we lay down the conditionto add jth column such thathj ̸= (uj−1hj−1 + uj−2hj−2 + · · ·+ uj−2bhj−2b) (4)+(vihi + vi+1hi+1 + · · ·+ vi+2bhi+2b),where ui, vi ∈ GF (q); i+2b < j−2b; if the coefficient ui (or vi) is non zero, then the othernon zero coefficients of ui (or vi) are confined to b consecutive positions of either side ofui (or vi), also if ui = 0 for (j − 1 ≤ i ≤ j − b), then ui = 0 ∀ i = j − 1 to j − 2b.This condition ensures that there shall not be a code vector which can be expressed assum (difference) of key errors of length b or less each. Thus, the codes so constructed willbe able to correct such errors.114 TWMS J. APP. ENG. MATH. V.5, N.1, 2015We now enumerate all possible linear combinations on the R.H.S. of (4):The coefficient ui on R.H.S. of (4) are such that if ui = 0 for j − 1 ≤ i ≤ j − b, thenui = 0 for j − b− 1 ≤ i ≤ j − 2b and if ui is non zero coefficient, then the other non zerocoefficients of ui are confined to b consecutive positions of either side of ui. The numberof such coefficient ui, including the zero vector, is given by(q − 1)qb+(q − 1)2qb + (q − 1)qb−1+(q − 1)2qb+1 + (q − 1)2qb−1 + (q − 1)qb−2+(q − 1)2qb+2 + (q − 1)2qb + (q − 1)2qb−2 + (q − 1)qb−3+........................................+.........................................+(q − 1)2q2b−2 + (q − 1)2q2b−4 + (q − 1)2q2b−6 + · · ·+ (q − 1)2q2 + (q − 1)q+1,which on simplification givesq2b+1 − qq + 1+ 1 =q2b+1 + 1q + 1. (5)To enumerate the coefficients vi is equivalent to the number of key errors of length b or lessin a (j− 1− 2b)-tuple. The number of key errors of length b or less in a (j− 1− 2b)-tuple,including the zero vector, is given by (refer Theorem 2.1)1 +q2b+1 − qq + 1+ (j − 1− 4b)(q − 1)(q2b+1 − qq + 1+ 1)+q2b+1 − q3(q + 1)2+(q − 1)(b+ q)q + 1. (6)Thus, the total number of linear combinations on the R.H.S. of (4) equalsexpr.(5)× expr.(6)=(q2b+1 + 1q + 1){1 +q2b+1 − qq + 1+ (j − 1− 4b)(q − 1)(q2b+1 − qq + 1+ 1)+q2b+1 − q3(q + 1)2+(q − 1)(b+ q)q + 1}. (7)Since these many linear combinations can not be equal to hj and at worst all the linearcombinations computed in (7) might yield a distinct sum, therefore in view of the factthat the total possible number of (n− k)-tuples is qn−k , the jth column hj can be addedto H provided thatqn−k > expr.(7). (8)To obtain a code of length n, we replace j by n in the above inequality and the inequality(8) becomesqn−k >(q2b+1 + 1q + 1){1 +q2b+1 − qq + 1+ (n− 1− 4b)(q − 1)(q2b+1 − qq + 1+ 1)+q2b+1 − q3(q + 1)2+(q − 1)(b+ q)q + 1}. (9)P. K. DAS: CODES CORRECTING KEY ERRORS 115Alternate Form 2.1 If N is the largest value of n satisfying the inequality (9), thenby replacing n by N + 1, the inequality in (9) gets reversed and we getqn−k ≤(q2b+1 + 1q + 1){1 +q2b+1 − qq + 1+ (N − 4b)(q − 1)(q2b+1 − qq + 1+ 1)+q2b+1 − q3(q + 1)2+(q − 1)(b+ q)q + 1}. (10)It should be noted that we don’t need to replace n in terms of N on the L.H.S. of aboveinequality since L.H.S. represents the number of cosets/vectors of length n − k whichremains the same.Alternate Form 2.2 If B is the largest value of b satisfying the inequality (9), thenfor b = B + 1, the inequality in (9) gets reversed and we getqn−k ≤(q2B+3 + 1q + 1){q2B+3 − qq + 1+ (n− 5− 4B)(q − 1)(q2B+3 − qq + 1+ 1)+q2B+3 − q3(q + 1)2+(q − 1)Bq + 1+ q}. (11)Example 2.1. For a (11, 2) linear code over GF (2), we construct the following 9 × 11parity check matrix H, according to the synthesis procedure given in the proof of Theorem2.2 by taking q = 2,b = 2 and n = 11.H =1 0 0 0 0 0 0 0 1 0 01 1 0 0 0 0 0 0 1 1 00 1 1 0 0 0 0 0 1 1 10 0 1 1 0 0 0 0 1 1 10 0 0 1 1 0 0 0 1 1 00 0 0 0 1 1 0 0 1 1 10 0 0 0 0 1 1 0 1 1 10 0 0 0 0 0 1 1 0 1 10 0 0 0 0 0 0 1 0 0 1It can be seen from Table 2.1 that the syndromes of any key error of length 2 or less areall nonzero and distinct. This shows that the code that is the null space of this matrixcan correct any key error of length 2 or less.Table 2.1Error pattern - syndromes tableError-patterns Syndromes Error-patterns Syndromes10000000000 110000000 00000111000 00000100111000000000 101000000 00010110000 00011101010100000000 111100000 00010101000 00011111111100000000 100100000 00010111000 00011100101000000000 011000000 00001101000 00001011101100000000 010100000 00001111000 00001000101010000000 011110000 00011101000 00010011101110000000 010010000 00011111000 000100001116 TWMS J. APP. ENG. MATH. V.5, N.1, 2015Cont...Error-patterns Syndromes Error-patterns Syndromes11010000000 101110000 00000010000 00000011011110000000 100010000 00000011000 00000010100100000000 001100000 00000010100 11111101000110000000 001010000 00000011100 11111100100101000000 001111000 00001011000 00001110100111000000 001001000 00001010100 11110001010110000000 111010000 00001011100 11110000110101000000 111111000 00000110100 11111011010111000000 111001000 00000111100 11111010101101000000 010111000 00001110100 11110111001111000000 010001000 00001111100 11110110111101000000 100111000 00000001000 00000001111111000000 100001000 00000001100 11111111100010000000 000110000 00000001010 01111110100011000000 000101000 00000001110 10000000100010100000 000111100 00000101100 11111001100011100000 000100100 00000101010 01111000101011000000 011101000 00000101110 10000110101010100000 011111100 00000011010 01111101101011100000 011100100 00000011110 10000011100110100000 001011100 00000111010 01111011100111100000 001000100 00000111110 10000101101110100000 010011100 00000000100 11111110001111100000 010000100 00000000110 10000001000001000000 000011000 00000000101 11001001100001100000 000010100 00000000111 10110110100001010000 000011110 00000010110 10000010000001110000 000010010 00000010101 11001010100101100000 001110100 00000010111 10110101100101010000 001111110 00000001101 11001000000101110000 001110010 00000001111 10110111000011010000 000101110 00000011101 11001011000011110000 000100010 00000011111 10110100000111010000 001001110 00000000010 01111111000111110000 001000010 00000000011 01001000100000100000 000001100 00000001011 01001001000000110000 000001010 00000000001 00110111100000101000 000001111P. K. DAS: CODES CORRECTING KEY ERRORS 1173. ConclusionThe paper presents lower and upper bound for a code correcting key errors of length b orless. The equality of the inequality in the statement of Theorem 2.1 (Lower bound) gives usthe optimal case and the corresponding codes. These codes are termed as optimal/perfectcodes as they correct key errors of length b or less and no others. These codes are usefulin communication due to their high rate of information. To study such optimal codesremains a further study.References[1] Das, P. K., (2014), Codes on Key Errors, Cybernetics and Information Technologies, 14(2), pp. 31-37.[2] Hamming, R. W., (1950), Error-detecting and error-correcting codes, Bell System Technical Journal,29, pp. 147-160.[3] Peterson, W. W. and Weldon(Jr.), E. J., (1972), Error-Correcting Codes, 2nd edition, The MIT Press,Mass.[4] Reiger, S. H., (1960), Codes for the Correction of Clustered Errors, IRE Trans. Inform. Theory, IT-6,pp. 16-21.[5] Sacks, G. E., (1958), Multiple error correction by means of parity-checks, IRE Trans. Inform. Theory,IT-4, pp. 145-147.[6] Sharma, B. D. and Gaur, A., (2013), Codes Correcting Limited Patterns of Random Errors UsingS-K Metric, Cybernetics and Information Technologies, 13(1), pp. 34-45.Pankaj Kumar Das for the photography and short autobiography, see TWMS J. App. Eng. Math.,V.3, N.2.

Codes correcting key errors

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Codes correcting key errors

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