On the capability of (T, U) permutation decoding method

~ 192 zyxwv zyx zyxwvutsrqponmlk zyxwvu zyxwvu IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. U314,FEBRUARYIMARCWAPRIL1994 On the Capability of (T, ,U)Permutation Deco ding Met 110d Ming Jia, Anader Renyamin-Seeyar and Tho Le-Ngoc Abstract- Error-trapping decoding techniques are attractive due to their simple structure. Since 1962 several improved error-trapping methods have been devised in an effort to extend the capability and effectiveness in decoding multipleerror-correcting cyclic codes. Prarige and MacWilliains introduced a (T, U ) permutation group applied to this errortrapping decoding strategy by making use of a set of codepreserving permutation to obtain k error-free positions from which the rest of the code word could be reconstructed. Recently, exact lower bounds on the code length n for (n, k, % + I ) cyclic codes have been found by using 2-step and 3step (T, U) permutation groups. This paper presents a study on the relationship between the code parameters n, k, t and the number of permutation steps s, with 1 being odd. Some examples on the capability of (T, U) perniutation decodable (PD)cyclic codes are illustrated. tions between code parameters n , k , t and the number of permutation steps, s, are also given. Some numerical results on the capability of ( T , U ) P D cyclic codes are also illustrated. zyxwvut zyxwvutsrqpo I. INTRODUCTION A decoder based on error-trapping decoding technique employs a very simple combinational logic circuit for error detection and correction [I], [2]. It is most effective for decoding single-error-correcting codes, some short-doubleerror-correcting codes, and burst-error-correcting codes. However, when it is applied to long and high rate codes with large error-correcting capability, it bec,omes very ineffective and much error-correcting capability will be sac,rificed . Since 1962, several improved error-trapping methods have been devised in an effort to extend the capability and effectiveness of the decoder for multiple-error-correcting cyclic codes [ 3 ] - [ 5 ] . Prange and MacWilliams have modified this decoding scheme by introducing a permutation group [4],[ 5 ] . Basically, the technique makes use of a set of code-preserving permutation to obtain IC error-free positions from which the rest of the codeword could be reconstructed. Recently, exact lower bounds on the code length 71 for ( 7 1 , k , 2t 1) cyclic codes have been found by using group (T, U ) permutations for 2-step and :%-step (T, U ) permutation decodable ( F D ) binary cyclic codes [6]-[8]. In this paper, we present a study on the capability of the general (T, U ) permutation method and the number of permutation steps used to achieve this capability with t being odd. Equations representing the exact rela- 11. (T, lJ) PERMUTATION For every cyclic code C in the vector space Fsupn of dimension n , with symbols from the finite field F = G F ( q ) , where q is the size of the field, there are various code preserving permutations. In this paper we specifically use the following group ( T , U ) permutation which is applicable to cyclic codes. Let (7 be a (71, k , t ) cyclic code over G F ( q ) If C ( X ) is a code polynomial, then e(.) can be represented as: n-1 j =0 where b j is a symbol from GF(q). The group (T, U ) permutation is defined as follows: zyxwvutsr zyxwvutsrq + P a p e r approved by Dariush Divsalar, t h e Editor for Coding Theory and Applications of t h e I E E E Communications Society. Manuscript received August 5 , 1991; revised September 3, 1002. T h i s paper has been partially presented at t h e Chnadian Conference on Electrical and C o m p u t e r Engineering, Quebec, C a n a d a , September 25-27, 1991. Ming Jia and T h o Le-Ngoc a r e with t h e Department of Electrical and Computer Engineering, Concordia Iiniversity, Montreal, Quebec, C a n a d a H3G 1M8; Anader Benyamin-Seeyar is with t h e Spar Aerospace Limited, Baie d’Urfe, Quebec, C a n a d a H9X 3T5. IEEE Log Number 9400890. and e”(.) = U%(.) n-1 zy where p is the characteristic of G F ( q ) , and the symbols of cyclic code C are from C F ( q ) . If p is relatively prime t o n , then the code is invariant under group ( U ) permutation [Ti] So, when p is relatively prime to n , is also a code word. Suppose ~ ( z= ) e(.) + e(z) is the received codeword polynomial where e(z), the error pattern polynomial, is of weight t or less, then the syndrome of the permuted received word sip(z) = ( 2 f i [ ~ ( z ) ] 2 m ’ od (zn 0090-6778/94$04.00 0 1994 IEEE + 1)) m o d g(z) zyx zyxwvutsrqponm zyxw zyxw zyxwvutsrqponm zyxwvutsr 193 IEEE TRANSACTIONS ON COMMUNICATIONS,VOL. 42, NO. 21314, FEBRUARYMARCWAPRU 1994 t is sip(") =3 I 2 7 t =5 zyxwvuts zyxwvuts zyxwv zyxwv = ( ~ ~ [ e ( c ) mod ] ~ ' ( P + 1)) mod g(5) We define e(.) to be a permutation decodable ( P D ) pattern if values of i and ,f3 exist such that e Z p ( x )= ( x p [ e ( x > l 2 ' mod (xn + 1)) * k o = 2 k has degree 71 - k - 1 or less. T h a t is, all the errors in the permuted e(z) are confined to the first I I - k parity-check positions and *$ip(x)= e z b e t a ( 2 ) . * +5 * k o = 2 k + t -2 * In this case the error pattern is s =s +1 + 1) (x-0 . .sip(z)>2-' m o d (xn s =s +I [6]. If these conditions on i and /I hold for every eip(x), such that the error e(z) is P D with i = 0, 1, . . . , Y. < y, where y is the least integer satisfies that 27 = 1 niod 7 1 , then we say that the code C is ( s 1)-step PD. + 111. MAIN RESULTS A . Code Rate and Information Length k For the ( T , U ) permutation decoding method, the relation between the code rate and k for the s-step permutation decodable ( P D ) codes is shown in Fig. 1: x k'n 1It 0 t Fig. 2 Procedures to solve for k* and s In the following we will present the procedures to determine k * , s and to establish the relation between n , le, and t . Proofs of these procedures are lengthy and therefore not included in this Transaction Letter. Part of the proofs is given in [SI. 7 1 , k and t When k 2 k* In the case of k 2 I%*, the ( n , k o , t o ) or ( 7 1 , k,, t o ) codes with C'. The Relatron of n ;L s-step I - I I ' k* Fig. 1. zyxwvut k Relation between and k where R' is mainly determined by t . When 1, 5 k * , the code rate for the s-step P D codes is around the value R'; but when k > IC*, will decrease monotonically and the code rate will approach +, which is the basic capability of error-trapping decoding technique. x B. The Turn Poznt k* The turn point k* is determined by the procedures shown in Fig. 2, which gives the k* and its corresponding number of permutation steps s . If the information leng1,h k is even, set ko = k , - 1 for this procedure. It is noted that k* is determined when t and s are given and k* is more than doubled for each additional permutation step. stop Fig. 3 t Procedures to solve for I zyxwv z zyxw zyxwvutsr 194 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 21314, FEBRUARYMARCWAPRU 1994 + 2 ) - 2“’(1+ 11 = t,(k, 71 = to(ke or 1) niod 2’-l, (4) where nkin is derived from (5) with k = k*. From (8), zyxwvuts + 1) - 2’p1(1 + 1) m o d 2’p1 (5) are s-step P D , where 1 is determined by the procediire shown in Fig. 3 for a given t o . The code. rate R’is determined by: I,. R‘ = * ?1(k*) where 11(k*) is the smallest integer which makes the c,ode ( 7 1 , k * , t ) s-step decodable. It is noted that although k* is a function of s , R‘ is mainly determined by t . D. The Relatzon of 7 1 , k and t W h m k < k* In the case of k < k * , the ( 7 1 , k,, t o ) codes with the following two conditions are .?-step P I ) : k 5 R‘ n (7) - k’ = > 7l +jas-l t (8) k where j is the smallest integer that makes k’ to be an integer. It can be seen from Fig. 1 that in the region of k < k * , the code rate for the s-step P D codes is around the value R’.Because R’is mainly determined by t o , so when k < k * , it will be very inefficient to make more permutation steps to increase the code rate of a s-step P D c.ode. E. Illustrative Exampless In this section, some examples are presented to illustrate the application of our results to search for P D cyclic codes. Example 1: In this example, we use the procedures given above to find out if the BCH (31, 16, 3 ) code is permutation decodable and what is the minimum number of permutation steps needed to decode this code. At first, we use the procedure given in Fig. 2 to find the rninirniim number of steps at which the turn point k* is larger than k,. This yields s = 5, k* = 21 > k, = 16. k, - 1 > - 71 = 0.4838709 zy zyxwvu zyxw 11 1 2 585 :3 585 4 5 585 575 559 527 6 7 463 8 335 zyxw zy R, = 6 0.33333 0.33333 0.33333 0.:33913 0.34884 0.37002 0.42117 0.58209 s 11 R5 - n 1 2 3 4 5 6 63 63 0.33333 0.33333 0.33333 63 53 37 37 0.39623 0.56757 0.56757 * Rs-1 -1 0% 0% 1.7% 2.9% 6.1% 13.8% 38.2% e - 1 0% 0% 18.9% 43% 0% (9) Then we verify the relationship between the code rate IC’ and R‘ as follows: 21 s VI. CONCLUSIONS From the above results, the following conclusions can be rn ad e : 1. The main goal of (T, U) permutation method is to increase k*. For each additional step, IC* will increase more than the double. When k 5 k * , the code rate for the s-step P D codes is around the value R’, which is much higher than the code rate for the basic error-trapping decoding technique +. zyxwvutsrqponmlk zyxwvutsrqp zyxwvutsrqp zyxwvutsr IEEE TRANSACTIONS ON COMMUNICATIONS,VOL 42, NO 21314, FEBRUARYNARCWAPRIL 1994 195 2. For given k and t , there exists an optimum number of steps which provides the largest improvement in code rates of P D codes. This occurs when k’ has just satisfied the condition k 5 k * . 3. R’ is mainly determined by t . So, after the permutation steps increase to this extent that satisfying k 5 k * , it will be very inefficient to make more permutation steps in order to increase the code rate of the P D codes. This is because the numbers 0, 1, . . . , T J - 1 can be partitioned into subsets which are invarj ant under U , and the union of any number of invariant subsets is also invariant under [ J . These subsets limit the capability of the permutation method. zyxwvu zyxwvu REFERENCES S. Liii aiid D. J. Costello, Error-Control Coding; Fundamentals and Applications, Prentice-Hall, Eiiglewood, New Jersey, 1983. M. E. Mitchell, “Coding and Decoding Operation Research,” [31 [41 [51 [71 t81 [91 E/ectronic Final Report on Contract AF 19(60Q)-6189, Air Force Cambridge Research Labs., Cambridge, Mass.,1961. T. Kasami, “A Decoding Procedure For Multiple-ErrorCorrection Cyclic Codes,” IEEE Trans. Inf. Theorgr, vol. IT-10, pp. 134-139, Apr. 1964. E. Praiige, “The use of Iiiforiiiatioii Sets in Decodiiig Cyclic Codes,” IEEE Trans. Inf. Theory, vol. IT-8, pp. 85-89, Sept. 1962. F. J. MacWilliams, “Periiiutatioii Decoding of Systeiiiatic Codes,” Bell Syst. Tech. J., vol. 43, Part I, pp. 485-505, Jaii. 1964. A. Beiiyaiifiii-Seeyar, S . S. Shiva, and V. K. Bliargava “Capability of the Error-Trapping Teclmique in Decodiiig Cyclic Codes,” IEEE Tran. on Info. Theory, Vol. IT-32, No. 2, pp. 166-180, March 1986. S. G. S. Sliiva, A. Benyamiii-Seeyar, aiid V. K. Rhargava, “Oii tlie Permutation-Decodability of Triple-Error-Correcting Codes,” in Proc, d l s t Annu. Allerton Gonf. on Gommonications, Control, and Computing, Illiiiois IJiiiv. , Oct. 1983. M. Jia, A. Beiiyaiiiiii-Seeyar, and T. Le-Ngoc, “!Exact Lower Bounds 011 tlie Code Length of Three-Step I’eriiiutatioiiDecodable Cyclic Codes,” 1991 IEEE International Symposium on Information Theory, Budapest, Hungary, Julie, 1991. M. Jia, A. Beiiyaiiiin-Seeyar, aiid T. Le-Ngoc, “On The Minimum Code Length of s-Step ( T , U ) Periiiutatioii Decodable Cyclic Codes,” Subiifitted to the 1993 IEEE International Symposium on Information Theory, Sail Antonio, Texas, Jaii. 18-22, 1993. zyxwvutsrqp