Concatenated group theoretic codes for binary asymmetric channels*

Transcription

1 Concatenated group theoretic codes for binary asymmetric channels* by SERBAN D. CONSTANTIN and T. R. N. RAO Southern Methodist University Dallas, Texas ABSTRACT A brief description of group theoretic codes is given and their suitability for binary asymmetric channels is exemplified. Previous research has shown the superiority in the information rate of the group theoretic codes over the existing codes for binary asymmetric channels and has left open the problems posed by the encoding/decoding procedures. The present paper introduces more sophisticated codes constructed from the already existing single I-error correcting group theoretic codes. The new class of codes, which we will refer to as concatenated group theoretic codes, will have improved encoding/decoding features while maintaining a high information rate comparable with that of equivalent length group theoretic codes. As their name indicates, a code of length 2n will be obtained by concatenating two sets of group theoretic codes of length n. INTRODUCTION Given an abelian group G of order n + 1, one can put in 1-1 correspondence the binarv vectors of length D with the linear combinations (with coefficients 0 or 1) of non-zero elements of G. The correspondence is rather intuitive and is given by: partition the set of 211 binary vectors V of length n into n + 1 disjoint classes of vectors V o, VI'..., V n The linear combinations corresponding to vectors of a class Vi' will sum up to ai, and vectors of each class will form a group theoretic code. Since one tries to optimize the number of codewords in the code, we shall look for the set Vi> having the most number of vectors (codewords) in it. If the best possible code obtainable by this method is desired, one must consider all abelian groups or order n+ I and look at the classes generated by each group and then select the largest such class. For an n as small as 10, one must use a computer in order to generate all classes of vectors generated by a group of order 11 or higher. The error correcting properties and a more detailed description of group theoretic codes can be found in Reference 1. For a comparison of Hamming code 4 and Kim and Freiman code 3 with the group theoretic codes of the same length the reader is referred to Table I. In this paper Zn will have the standard meaning of the addition modulo n group. MATHEMATICAL CONSIDERATIONS Consider two vectors: n 2 lxi ai~ IXl' 1X2', IXn) i=l ( 1.1) (2.1) ai =l=a o are the non-zero elements of the group and Without loss of generality, assume the group operation to be addition. Moreover, the above correspondence will * The authors' research was supported in part by a grant from National Science Foundation Eng ai' b i i = 1,..., n are real numbers. Without loss of generality let's assume: at ~a2 ~a3:5... :5a n { b t :5b 2 :5b 3 :5..:5b n (2.2) If 1T is a permutation, 1T=(il' i 2,..., in), then by 1T(B) we will denote 837

2 838 National Computer Conference, 1977 TABLE I-Number of Codewords for Single-Error Correcting Codes Group Group Code Theoretic Theoretic Length Kim-Freiman Codes Codes n Hamming Code Code G=Zn+lt ** =4 4* =4 22+2= = = 12 10* = = = = = = = =80 94* 94 II 2 7 = = = = = = = = = = = = = = = = = = = = * Figures in these entries are best possible using this method, due to the fact that there is only one group of order n+ I. See Reference 5 for the number of groups of various order. ** The groups considered were the additive groups of the Galois fields GF(pq). Note: The number of codewords in the group theoretic codes in the above table was generated by computer. t These codes were also obtained by Varshamor. 6 is: This follows immediately from: (a 2 -a 1 )(b 2 -b 1 );===0 by expanding the product. Having proved this, to get the general result we are only one step away. Consider any two permutations 7Tl and 7T2 and assume: M(A, B)=7Tl(A)'7T2(B) (2.5) (2.6) and let p be the largest integer such that the scalar product 7Tl(A)'7T2(B) does not contain ap bp as a term. Then '7Tt(.ii1 )-u2(b), must certainly contain ap;bi+afbp But then using- the trivial case considered above, i.e., substituting ap'bj+aj"bp in the scalar product 7T1(A)'7T2(B) with aj"bj+ap'bp we contradict that M(A, B)=7Tl(A) 7T2(B). Repeating the procedure described until p= 1 we obtain the desired result. Q.E.D. Corollary I: For any arbitrary vector A=(a 1, a2,...,an) i.e., the vector whose elements are the elements of B permuted according to 7T. We are interested in finding 7T* such that A B 7T :s;a B 7T * holds for any other permutation 7T, by A B is meant the scalar product of the two vectors i.e., II A'B= " a b,,(.. I I (2.3) j=1 This maximum scalar product i.e., A'B 7T * will be referred to M(A, B), M acts like an operator on the two vectors A and B into the real numbers. Lemma I For any two vectors A and B satisfying (2.2): Proof M(A'B)=A'B=B'A (2.4) Let's consider first the trivial case, i.e., II M(A, A)=A'A= L aj 2 (2.7) 1=1 Similarly, if we define the operator mea, B) to be: for any permutation 7T, then the following similar result is obtained: Lemma 2 For any two vectors A and B satisfying (2.2): Proof' mea, B)= L aj'bn- i + 1 i=1 II (2.9) Similar to the one for Lemma 1. If the elements of a vector A are non-positive and the elements of a vector B are non-negative the following relations hold: such that What we have to show then to prove the lemma in this case M(A, B)=-m(IAI' B) { mea, B)=-M( A,B) (2.10)

3 Binary Asymmetric Channels 839 Finally, one more result is needed. We want to construct a single error correcting code of length two, over the alphabet {O, 1, 2,..., n}, the only possible single errors occurring in a codeword (i, j) are: { (i,j)~(i-1,j) i*o (2.11) (i, j)~(i, j -1) j *0 Maximizing the number of codewords in such a code is also one of the objectives. Consider the following picture: the third property, i. e., no other code can have more codewords than the code described above. Obviously, the number of possible 2-tuples over the alphabet {O, 1, 2,..., n} is (n+1)2. Let n+1=3m+k, k=(n+ 1) mod 3. Then, the number of codewords in our code is: m (n+ 1)+ L 2'[(n+ 1)-3'i]=3'm 2 +2'm'k+k (2.12) i=l In general, for every codeword (i, j) that we include in our code, we eliminate from the list of potential codewords two other tuples, (i -1, j) and (i, j -1); i.e., the contaminated tuples corresponding to the codeword (i, j). With this observation, one can actually convert the original problem into a tile covering problem, namely we will be concerned with covering an (n+1)x(n+1) rectangle of squares with tiles of the shape and orientation of the one in Figure 2, such that no two tiles overlap and as much as possible of the surface of the (n + 1) x (n + 1) rectangle is covered with tiles. Figure 2 Figure 1 Let (i, j) be the name of the square in the ith row and jth column. The collection of the (i, j) 2-tuples corresponding to the shadowed squares in Figure 1, can easily be checked to form a single error correcting code under the conditions of error occurrence as described by (2.11). Note that no additional square could be shadowed such that the augmented code be still single error correcting. However, this does not pr()ve that no other code could have more codew~rds than' this code... Lemma 4 Consider the following length 2 code over the alphabet {O, 1,2,..., n}: (a) (i, i) is a codeword for i=o, 1, 2,..., n (b) if (i, j) is a codeword, so is (i+ 3, j) and (i, j+ 3) for i+3:5n andj+3:5n Then, such a code is cyclic, is single error correcting and contains the maximum number of codewords. Proof. The first two assertions can be easily disposed of using the definition of the code and (2.11). We shall prove only The simplest upper bound for the number of tiles covering the rectangle is b f t 'l -< [area of the SUrfaceJ _ [(n+ 1)2J max num er 0 I esarea f' 1 t'l a smg e I e because = (3m+k)2J [k2j [--3- =3m 2 +2mk+ "3 =3m 2 +2mk+k But, the upper bound coincides with the number of codewords in our code. Q.E.D. The method of finding such codes can be generalized in different directions. For example, one can build a single error correcting length 2 code the first component can take on values from {O, 1,..., n} and the second component can take on values from {O, 1,..., m}. The code could be constructed using a similar picture as the one on Figure 1 except that the surface will be an (n+1)x(m+l) rectangle. Lemma 5 Given N=m+n, N fixed, the length 2 code with the most number of codewords, is obtained for n=n/2, m=n/2.

4 840 National Computer Conference, 1977 Proof. Area of the (n + 1) x (m + 1) rectangle is maximum when n=m. Q.E.D. U sing extensions of the rules given in Lemma 4, one can build a single error correcting length k code over the alphabet {O, 1,..., n}. Proving the maximality of the number of codewords of the length k code constructed by the rules similar to the ones in Lemma 4 requires more involved calculations. At last, the length 2 code over the alphabet {O, 1,..., n} will be called a weight-code and will be used for constructing the concatenated group theoretic codes. CONSTRUCTION OF CONCATENATED GROVP THEORETIC CODES Let GI and G2 be two abelian groups of the same order n+ 1 (as it will be seen GI and G2 need not necessarily be distinct) and let {Yo, VI'..., Vn} and {Uo, UI,..., Un} be the partitions of the set of 2 n binary vectors of length n into n + I disjoint classes of codes as induced by the two groups G I and G2 respectively. Each class (code) Vi' Vi i=o, 1, 2,..., n is a group theoretic code in itself. Distinguishing the codewords in each class by their weight (i.e., # of ] 's in the codewords) we obtain the following classification TABLE II 2 n Vo aoo aoi '. aon VI Vn a no ani,. ann Vo VI V n bno TABLE III 2... n boo bol,... bon bnl,..:... bnn ajj = # of vectors (codewords) of weight j in the class Vi bij = # of vectors (code words) of weight j in the class Vi The length 2n code we will construct, as its name suggests, will be the result of concatenating codewords from some class Vi with codewords from some class Vj in a manner that will result in a maximum number of codewords of length 2n. For ease of reference, let's adopt the following notations: C the concatenated group theoretic code to be constructed W the weight-code. This is a length 2 code, over the alphabet {O, 1,..., n} as described earlier. C I the set of codes generated by G I i.e.: V 0, VI"'" Vn C2 the set of codes generated by G2 i.e.: Vo, VI,..., Un lower case letters will be used to denote codewords I w I weight of codeword w. Then, a condensed description of C could be given in the following form: C={(WIW2) IIwII =i, IW21 (3.1) =j, (i,j)ew, WIEVk, W2EUI) aki and bij are both the rth largest elements, for some r, in the ith column of Table II and jth column of Table III, respectively} For the case GI and G2 have been selected to be one and the same group, in the above definition (3.1) pick k=l. (See Corollary 1). Given the two groups GI and G2 of order n+ 1, one selects a length 2 weight-code, as described in Lemma 4, determining what weights the codewords to be paired together should have. Let Ci' denote the ith column of Table II and ct denote the jth column of Table III, i, j=o, I, 2,..., n. Then, for every codeword (i, j) of the weight-code W, we will generate codewords (WIW2) of C, WI is a codeword in CI of weight i corresponding to some code V k and W2 is a codeword in C2 of weight j corresponding to some code V I k and I are selected as dictated by Lemma 1 applied to the vectors C/ and C/,. Error detection and correction in C: Let (rlr2) be a received message and let's assume that at most one I-error has occurred to the transmitted message (WIW2) of C. From the construction of C we know (Iwll, IW21)EW. Let i= Irll and j= Ir21. If (i, j) is a valid codeword of W then (rlr2)=(wiw2) and hence no error has occurred. However, if (i, j)f/=w then a single I-error has occurred to the transmitted message (WIW2) and more precisely a single 1- error has occurred to either WI or W2 decreasing the weight of one of the two codewords by 1. Therefore, either i= IWII and j= IW21-I; i.e. the error has occurred in W2 or i = I WI 1-1 and j = I w 21; i. e. the error has occurred in WI' Since W is a single error correcting code we can determine(lwll, IW21> and hence know the single I-error has occurred. Let's suppose the error has occurred in W2 and as a result we have received (rlr2)=(wlr2). By the correspondence (1.1) and the structure of GI we establish the membership of WI in some class (code) V k' Let r be the rank of aki in Ci'. Find then the rth largest element (the element of rank r) in CHI" and let this be b lo +l)' Now we know that r2 must have come from a codeword of VI and since V I is a single ] -error correcting code we can correct r 2 and obtain w2. Thus, we will correct (r 1 r2) and obtain the transmitted message (WIW2) of C. The following example goes through each step of the

5 Binary Asymmetric Channeis 841 detecting and correcting procedure described in the previous paragraph for a length 16 concatenated group theoretic code generated by the additive group of the Galois field GF(32). Example Consider the additive group G9 of the Galois field GF(3 2 ) whose addition table is given below: X X I+X X X X I+X 2+X 2X 2X 2X I+2X 2+2X 0 I+X I+X 2+X X 1+2X 2+X 2+X X I+X 2+2X 1+2X i+2x 2+2X 2X X 2+2X 2X 1+2X 2 2X I+X 2+X 1+2X 2+2X 2X l+x 2+X 1+2X 2+2X I+2X 2+X X 2+2X 2X 2+2X X I+X 2X 1+2X 0 1+2X 2+2X 1 2 X 1 2 I+X 2+X 1 2+2X 2X X 1+2X 0 1 l+x X X 2+X 0 1 X I+X For each i, i=o, I,...,8 the codewords in Vi form a group theoretic code, and for purposes of clarity, the codewords of such a code, namely of Vo, will be listed out. They are: co=(o,o,o,o,o,o,o,o); c3=(0,0,0,0, 1,0,0, 1); c6=( 1,1,0,0,1,0,0,1); c9=(0,0, 1,1,0,1,1,0); CI2 =(1, 1, 1, 1,0, 1, 1,0); C 15 =( 1,1,1,1, 1, 1, 1, 1); c I8=(0,1,1,0,0,0,1,0); C21 =(0,1,0,0,0,1,0, I); C24=(1,1,1,0,1,1,0,0); C27=(0, 1,1,1,0,1,0,1); c 30 =( 1,0,1,0,0,1,1,1); c1 =( 1,1,0,0,0,0,0,0); c4=(0,0,0,0,0, 1,1,0); c7=(1, 1,0,0,0, 1,1,0); ClO=(O,O,O,O, 1,1,1,1); C 13 =( 1, 1,0,0, 1, 1, 1, 1); c I6=(1,0, 1,0,0,0,0, 1); CI9=( 1,0,0, 1,0, 1,0,0); C22=(0'0, 1,0, 1,1,0,0); c 25=(I, 1,0, 1,0,0,1,1); C28=(0, 1,1,0,1,0,1,1); C31 =(0,1,0,1,1,1,1,0) c2=(0,0, 1,1,0,0,0,0) c5=(1, 1,1,1,0,0,0,0) c8=(0,0, 1,1,1,0,0,1) C ll =(1,1,1,1,1,0,0,1) c 14 =(O,O, 1,1,1,1,1, I) CI7=(0' 1,0, 1,1,0,0,0) c 20=( 1,0,0,0,1,0,1,0) C23=(0,0,0, 1,0,0, 1,1) C26=(1,0, I, 1, 1,0, 1,0) C29=( 1,0,0, 1,1,1,0,1) The distribution by weight of the codewords in the nine classes V 0' VI'..., V 8 induced by G9 into the set of 2 8 binary vectors of length 8 is given by the following table: TABLE IV ~Weight class ~ df 1 U u I! The length 16 concatenated group theoretic code C will be constructed from C1, the set of codes generated by G1=G9; and C2, the set of codes generated by G2=G9. In this case G1 and G 2 will be identical and equal to the addition group of GF(3 2 ). This implies that C1 and C2 will be identical and equal to the set of codes V 0, VI'..., V n. The weightcode W will be the one determined by the names of the shadowed squares of the picture in Figure 1. To each codeword (W1W2) of C it will correspond a codeword (\w1\, \w2\) of Wand conversely to each codeword (i, j) of W there will correspond a set of codewords in C. The number of codewords in C corresponding to a codewoui 'i..il ill.w is giveniu the. fquo,w.ing t,abl~: \w 1\ \w2\ TABLE V i I

6 842 National Computer Conference, 1977 Let and (3.3) (Vji'Vkl)={(VIV2) IvlEVji, V2EVkl} Then, using the notation (3.3), the set of codewords (WIW2) of C corresponding to the codeword (2, 5) of W is: {(V 0 2 V0 5 ), (V/'VI 5 ), (V2 2 V2 5),., (Vl'V 8 5 )} which comprises a total of 176 codewords. To see how detection and correction is done when a single I-error has occurred, let: r=(r Ir2)=( l) be the received message and assume (WIW2) of C was the actually transmitted message. For the above received message we have: i= Irll= 1( ) 1=4,j= Ir21= 1( )1=3 and since (4, 3) is not a codeword of Wand assuming a single I-error has occurred, the codeword of W corresponding to the transmitted message should have been (4, 4). Thus, r2 is in error. Since it must be that rl =( ) EV 3 1 V 3 41 = 8 and 8 is the largest element in the 4th column of Table IV. Since there are seven other entries equal to 8, suppose the following pairing of codewords of C 1 and C2 corresponding to the codeword (4, 4) of W, has been done: (except for (Vo 4. Vo 4) all the other pairings are done arbitrarily; yet one must know before trying to correct transmitted messages what the pairings are). Therefore, rl must have been paired with a codeword of weight 4 from V 6' However and the error has occurred in the third position as given by Thus the transmitted message was (WIW 2 )= (lo ). The number of code words in C as given in Table V is 2470 and comparing it with the Kim & Freiman code of length 16 or the Hamming code of the same length (see Table I) we see a definite improvement in the information rate. Although the number of codewords in C is slightly less than the number of code words in the group theoretic code of length 16, efficiency could be gained in the encoding/ decoding process. If ROM was to be used for encoding and decoding purposes, smaller size ROM could be utilized if a message m=(m 1 m 2 ) was encoded as: and decoded as: (mlm2)~{~~:~} ~WIW2)EC Longer weight-codes could be used in generating concatenated group theoretic codes, but it appears as though this would have a negative impact on the encoding/decoding procedures and maybe in the correction process while accomplishing a high information rate. An optimal concatenation (pairing) of three or more smaller codewords into a longer code is also not apparent, and could be regarded as a natural generalization of the method presented in this paper. CONCLUSIONS The future research to be pursued by the authors of this paper will be focused in the direction of finding efficient encoding/decoding procedures for both group theoretic codes as well as concatenated group theoretic codes. Finding an efficient encoding/decoding procedure for the two types of codes is believed to be possible due to the structure of the two codes inherited from the groups that have generated them. REFERENCES I. Constantin, Serban D. and T. R. N. Rao, "Group Theoretic Codes for Binary Asymmetric Channels," Technical Report CS 76014, Department of Computer Science, Southern Methodist University, Dallas, TX. 2. Rao, T. R. N. and A. S. Chawla, "Asymmetric Error Codes for Some LSI Semiconductor Memories," 7th Annual Southeastern Symposium on System Theory, March pp Kim, Wan H. and Charles V. Freiman, "Single Error Correcting Codes for Asymmetric Channels," I.R.E. Transactions on Information Theory, June Peterson, W. W. and E. J. Weldon, Jr., Error Correcting Codes, M.I.T. Press, Cambridge, Massachusetts, Hall, Marshall Jr., The Theory of Groups, Macmillan, Varshamor, R. R., "A Class of Codes for Asymmetric Channels and a Problem from the Additive Theory of Numbers," Trans. on Information Theory, January 1973, pp