+ a anhn. h 2

Transcription

1 NEW MAJORITY-LOGIC DECODABLE CODES By C. L. CHEN Summary.-A method of the construction of majority-logic decodable codes is presented. Many of the codes constructed are more efficient than any other comparable majority-logic decodable codes previously known. Introduction.-Majority-logic decodable codes are attractive for errorcontrol in data transmission because of their simplicity in implementation. Since Massey introduced the concept of orthogonal parity-check sums in 963 and Rudolph constructed the class of finite geometry codes in 964, a lot of effort has been expended on the construction of majority-logic decodable codes and their decoding schemes. Many important results have been obtained [ -6] In general the decoding of finite geometry codes is considered to be simpler than that of other types of codes. However, the efficiency of finite geometry codes is generally lower than the well-known Bose Chaudhuri-Hocquenghem codes. Thus the construction of efficient majoritylogic decodable codes is an important problem. Recently several authors have generalized the construction of finite geometry codes [6-9], The generalized codes are majority-logic decodable, In addition, many of these codes are more efficient than the original finite geometry codes. In this paper the construction method of linear codes proposed in [o] is used to construct new majority-logic decodable binary codes. Many of these codes constructed are more efficient than any other comparable majority-logic decodable codes previously found including the generalized finite geometry codes. For examples, the new code of length 8 with 69 information digits and a majority-logic decodable distance of 6 has five more information digits than the best majority-logic decodable codes previously found with the same code length and distance, and the new code of length 5 with 44 information digits and a majority-logic decodable distance of 6 has 9 more information digits than the best comparable majority-logic decodable codes previously known. Some of the codes constructed are shown in a table. It is also shown that a special class of the new codes are cyclic codes. Throughout this paper binary codes will be assumed. The readers are assumed to be familiar with linear codes and cyclic codes. Majority-Logic Decoding.-A binary (n,k) linear code of length n with k information digits is a k-dimensional subspace of an n-dimensional vector space over the binary field GF(). Let~= (a,a,,an) be a transmitted code word, E = (r,r,,rn) be a received vector. Define! = (e,e,,en) as the error vector. Then E = ~ +! Leth= (h,h,,hn) be a vector in the null space of the code space. Then the scalar product of~ and his a h a h + a h anhn This work was supported in part by NSF Grant GK-4879 and in part by the Joint services Electronics Program under contract DAAB-7-7-C-59. Dr. Chen is with the Coordinated Science Laboratory, University of Illinois.

2 since a is a code word. r h The scalar product of rand his (~ +!).!; e h () Thus a vector h of the null space of a code forms a parity-check sum that checks on the sum of the values of the error digits at those positions corresponding to the nonzero components of h. Let I be the set of integers corresponding to the nonzero components of h. Since each component of his either zero or one, the parity-check sum of () can be written ass = _E.h. = z.:: e. (la) Now suppose that sums can be formed by a set of J vectors in for a binary linear code the following J taking the scalar products of a received the null space. sl z.:: e. + z.:: e.. Sz z.:: e. + z.:: e. parity-check vector with SJ z.:: ei + z.:: e. J () where the integers in I,I,,IJ are all distinct. Notice that ei appears in every check sum for, and ei appears in at most one of the J check sums for ifi The set of the J check sums is said to be orthogonal on the sums' = Z.:: e. If the number of errors in the received vector is. less than or equal to J/, then the value of s' can be assumed to be if more than J/ of the sts are, and assumed to be O otherwise. Once the value of s' is obtained, s' can be considered a new parity check sum and added to the set of the existing parity check sums. If a set of J check sums orthogonal on a new sum, say S, can be formed from the enlarged set of parity-check sums, then the value of Scan be determined by the majoritylogic decision rule described above. Again, S becomes a new check sum and is added to the set of the existing check sums. If this procedure can be continued until a set of J check sums orthogonal on a single error digit is formed, then the value of the error digit can be determined. The code is said ~o be majority-logic decodable with majority decodable distance d = J+l if every error digit can be determined in this way. Note that d may not be the true minimum distance of the code. In the following sections (n,k,d) denotes a binary linear code of length n with k information digits and a majority-logic decodable distance of d. Construction of New Majority-logic Decodable Codes.-Let c be an (n,k,d ) majority-logic decodable code and c be an (n,k,d ) majoritylogic decodable code. The new code C is defined by C = [zjz = (x,x+y), xec, yecz} The length of code C is n, and the number of information digits is k +k In the following it is shown that code C can be majority logic decoded up to distanced= min(dz,d) Thus d = dz if dz= d. The generator matrix G of code C can be depicted as follows

3 G where G and G are the generator matrices of c and c, respectively. A row vector in G is either a vector of G repeated once or a vector of G. Let H and H be the parity check matrices of c and C respectively. Then the parity check matrix Hof code C is H = Let R = (r,r,,rn,r,,rn) = R+R be a received vector corresponding to the input code word Z = (x,x+y), where R = (r rn), R = (r rn), and xec, yec The error vector E = (e eln e e n) = R-Z can be expressed as E = (E,E ), where E = (e en), E = (e,,e n) Thus R = x+e and R = x+y+e For a vector in H there is a vector ii\ H which is the vector in H repeated once. Thus for a parity check sum S = ifrei in code c, there is a parity check sum S' = ~ (e.+e.) in codec. Therefore, if a set of J = d-l orthogonal check sums on Scan be formed in c, a set of J orthogonal check sums on s' can be formed inc. Following the majoritylogic decoding procedure for code C, the values of (e i+e i), i=l,,,n can be determined provided that the number of errors is less than_.r equal to J/. Since H is a submatrix of H, a set of J orthogonal check sums can be formed in each decoding step for code C. Suppose that the following set of J orthogonal check sums is formed. sl ~ e + S ~ e + ~ e ~ e S3l ~ e + ~ e ier li J Since (ei+ei) has been determined for i orthogonal check sums can be formed. s' ~ e + ~ e. 8 I ~ e + ~ e. sl + L (e+e.) 8 I L e. + L e. ier. S, =.L eli + ~ e. Ji ei ie 3,,n, the following J + The set of J + check sums in (3) and (4) are orthogonal on i~i e i. The value of L e. is determined according to the usual majority-8gic decision rulg. Following the decoding procedure for code C, a set of (3) (4)

4 J + orthogonal check sums can always be formed. Thus, the values of eli i =,,n can be determined provided that the number of errors is less than or equal to min((j /),(J+/)). Once e i is determined, e i can be determined from the check sum e i+ei for i =,,,n. Therefore-, code C is majority-logic decodable up to distanced= min(d,d ). Some new majority-logic decodable codes of length 8 and 5 constructed are shown in the Table. For those codes that have ID)re information digits than comparable codes previously discovered the difference is shown in the last column. For example, (8,3,3) code has ID)re information digits than the (8,9,3) Reed-Muller code. TABLE. Some new majo~ity-logic decodable codes. Remarks C n k d C k d C n k d 64,4,6 7,3 8,3,3 64,37, 3, 8,5, +c 64,45,8,6 8,67,6 +3c 64,45,8 4,6 8,69, ,57,4 45,8 8,,8 +3c 56,45,64 9,8 5,54,8 +Sc 56,94,3 45,64 5,4,64 C 56,9,6 95,3 5,86,3 + 56,9,6 93,3 5,84,3 +loc 56,3,8 9,6 5,44,6 +9c 56,47,4 3,8 5,47,8 C Cyclic Codes.-If c and C codes are cyclic codes, and c is a subcode of c, then code C constructed in the last section is also a cyclic code. A cyclic code is specified by the roots of its generator polynomia Let Q,Ql,,Qls be the set of roots of the generator polynomial g (x) of code c, and Ql,Q,,QIJ, be the set of roots of the generator polynomial g (x) of code C, where J,~s, and QI~=. That is, g (x)=(x-a)(x"') (X-Ct's), and g(x)=(x-a )(x-' ) (x-a,). Then the parity check matrix of code C is Qll ~ H QI s QI s n QI s The columns of H can be permuted to become the following matrix:

5 H' Q' s n Q' s 3 Q' s Now consider the null space c' of H'. Let (a,a,a,,a n-l) be a vector of C'. Then and i =,,.,s i =,,.,f, Thus ',',,O's are roots of the polynomial a(x) = a +a x+a x +.. +a n_ x n-l corresponding to the vector (a,a,..,a n-l), and ',',...,O'i, are roots of a x+a 3 x +.. +a n_ x n-l = a'(x)x, where a'(x) is the formal derivative of a(x). In oth~r words, ',',,.,O';, are repeated roots of a(x). Therefore, a polynomi~l in the null space of H' is a multiple of the polynomial g(x)=(x-a) (x-a) (x-a;,) (x-a;,+) (x-as). The null space of H' is a cyclic code of length n whose generator polynomial is g(x). This code is equivalent to the original code C. Most of the codes shown in the Table can be put into cyclic form. If codes C and C are punctured with digits, they become cyclic codes with distance one less than the original distance. If C is a subcode of c, then C is a cyclic code. For example, let C be the (55,3,7) cyclic code and C be the (55,9,5) cyclic code. Then C is a (5,44,4) cyclic code. Those codes in the Table that can be put into cyclic form are marked with c in the last column. Conclusions.-Using the construction method in [o] we have constructed new majority-logic decodable codes. Many of these codes constructed are more efficient than the best comparable codes previously discovered. In addition, many of the new codes constructed are cyclic codes. The results reported in this paper should encourage one to search further for more efficient majority-logic decodable codes. There may exist majority-logic decodable codes with efficiency competitive with BCH codes. References [] J. L. Massey, Threshold Decoding, MIT Press, Mass., 963. [] L. D. Rudolph, "Geometric Configuration and Majority-Logic Decodable Codes," MEE Thesis, University of Oklahoma, Norman, Oklahoma, 964.

6 [3] E. J. Weldon, Jr., ''Euclidean Geometry Cyclic Codes," Proc. of Symeosium of Combinatorial Mathematics at the University of North Carolina, Chapel Hill, 967. [4] E. J. Weldon, Jr., "Some Results on Majority-Logic Decoding," Chapter 8, Error Correcting Codes, edited by Mann, John Wiley and Sons, N.Y., 968. [ 5] E. J. Weldon, Jr., "New Generalizations of the Reed-Muller Codes - Part II," IEEE Trans., IT-4, 99-6, 968. [6] T. Kasami, s. Lin and w. w. Peterson, "New Generalizations of the Reed-Muller Codes - Part I," IEEE Trans., IT-4, 89-99, 968. [ 7] D. K. Chow, "A Geometric Approach to Coding Theory with Application to Information Retrieval," Report R-368, CSL, Univ. of Illinois, Urbana, 967. [8] J. M, Goethals and P, Delsarte, "On a Class of Majority-Logic Decodable Cyclic Codes," IEEE Trans., IT-4, 8-89, 968. [9] K.J.C. Smith, ''Majority Decodable Codes Derived from Finite Geometries," Memo No. 56, Dept. of Statist. Inst., Univ. of North Carolina, Chapel Hill, 967. [] P. Delsarte, J.M. Goethals andf. J. Macwilliams, "On Generalized Reed-Muller Codes and Their Relatives," Information and Control, Vo 6, No. 5, 43-44, 97. [ ] T. Kasami and S. Lin, "On Majority-Logic Decoding for Duals of Primitive Polynomial Codes," IEEE Trans., IT-7, 33-33, 97. [] w. w. Peterson and E. J. Weldon, Jr., Error-Correcting Codes, Edition II, MIT Press, Mass., 97. [ 3] R. L. Townsend and E. J. Weldon, Jr., "Self-Orthogonal Quasi-Cyclic Codes," IEEE Trans., IT-3, 83-95, 967. [4] C. L. Chen, "Note on Majority-Logic Decoding of Finite Geometry Codes," IEEE Trans., IT-8, , 97. [5] L. D. Rudolph and w. E. Robbins, "One-Step Weighted-Majority Decoding, IEEE Trans., IT-8, , 97. [6] L. D. Rudolph and C.R.P. Hartmann, ''Decoding by Sequential Code Reduction," Report to be published, Syracuse University, 97. [7] P. Delsarte, "A Geometric Approach to a Class of Cyclic Codes," Journal of Combinatorial Theory, 6, , 969. [8] s. Lin and E. J. Weldon, "Multifold Euclidean Geometry Codes," Report to be published, University of Hawaii, 97. [ 9] C.R.P. Hartmann and L. D. Rudolph, "Generalized Finite-Geometry Codes," Proc. of the Tenth Allerton Conference on Circuit and Systems Theory, Univ. of Illinois, 97. [] N.J.A. Sloane and D, S. Whitehead, "New Family of Single-Error Correcting Codes," IEEE Trans., IT-6, No. 6, pp , November 97.