IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 14, NO. 6, NOVEMBER 1968 807 [41 [51 161 171 [81 methods in pattern classification,” IBM J. Research and Develop. vol. 8, pp. 299-307, July 1964. W. H. Highleymen, “A note on linear separation,” IRE Trans. Electronic Computers (Correspondence), vol. EC-lo, pp. 777-778, December 1961. H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proc. .%ndBerkeley Symp. on Mathematical Statistics and Proba- bility. Berkeley,. Calif. : University of California Press. 0. L. Mangasarran, “Linear and nonlinear separat,ion of patterns by linear programming,” Operations Research, pp. 444-452, Mav-June 1965. R. “C. Minnick, “Linear-input logic,” IRE Trans. Electronic Computers, vol. EC-lo, pp. 6-16, March 1961. T. S. Motzkin and I. J. Schoenberg, “The relaxation method for PI 1101 [ill linear inequalities,” Canadian J. Math., vol. 6, no. 3, pp. 393- 404, 1954. S. Muroga, “Logical elements on majority decision principle and complexity of their circuit,” presented at the 1959 Internat’l Conf. on Information Processing, Paris. S. Muroga, I. Toda, and S. Takasu, “Theory of majorit,y deci- sion elements,” J. Franklin Institute, vol. 271, pp. 376-418, May 1961.6 J. B. Rosen, “Iterative solution of nonlinear optimal control ~;r&lems,” J. SIAM on Control, vol. 4, pp. 223-244, February I am indebted to Dr. William C. Mow for this reference. Polynomial Codes TADAO KASAMI, MEMBER, IEEE, SHU LIN, MEMBER, IEEE, AND W. WESLEY PETERSON, FELLOW, IEEE Abstract-A class of cyclic codes is introduced by a polynomial approach that is an extension of the Mattson-Solomon method and of the Muller method. This class of codes contains several important classes of codes as subclasses, namely, BCH codes, Reed-Solomon codes, generalized primitive Reed-Muller codes, and finite geometry codes. Certain fundamental properties of this class of codes are derived. Some subclasses are shown to be majority-logic decodable. I. INTRODUCTION HIS PAPER presents a class of cyclic codes and their dual codes that contains many well-known classes of codes as subclasses, such as BCH codes [l], [a], Reed-Solomon codes [a], [3], generalized Reed- Muller codes [4], [5], projective geometry codes [6], [7], [8], and Euclidean geometry codes [6], [9]. This class of codes is introduced by a polynomial approach that is an ex- tension of the Mattson-Solomon method [lo] and the Muller method [ll]. The generator polynomial of any code in this class is characterized, and the minimum dis- tance is lower bounded. A code in this class is proved to be either a BCH code or a subcode of a BCH code of the same minimum distance. For some subclasses of this class of codes, the exact minimum distance is established. Some useful geometric properties are also proved. Because of the geometric structure, certain subclasses of this class of codes are shown to be majority-logic decodable. II. PRELIMINARIES Let GF(q”“) be the extension field of GF(q”), where m and s are positive integers and q is a power of prime. Let (I: be a primitive element of GF(q”“). Then ( a’, LY’, a’, . . . , ame1 ) Manuscript received December 8, 1967; revised March 11, 1968. This work was supported in part by the USAF Cambridge Research Laboratory, Office of Aerospace Research, Contract AF 19(628)- 4379. T. Kasami is with the Department of Control Engineering, Faculty of Engineering Science, OsakaUniversity, Toyonaka, Osaka, Japan. S. Lin and W. W. Peterson are with the Department of Electrical Engineering, University of Hawaii, Honolulu, Hawaii 96822 form a basis of GF(q”‘) over GF(q”). Any nonzero element LY” in GF(q”‘) can be expressed as ai = z aiiai-’ for 0 _< j 5 q”” - 2 0) where aii E GF(q’). For convenience, we shall call the m- tuple (ali, aZj, . . . , a,,.) as the coordinate vector of (Y’. It is known that q”” - 1 is divisible by q8 - 1. Suppose that b is a factor of q’ - 1. Let 2 = (q” - 1)/b, and (2) n = (q”” - 1)/b. Let X,, X,, . .. , X, be m variables a = (X,, *. * L X,). We define P(m, e, over GF(q”) and p, b) as the set of polynomials f(X) = f(X,, X,, * * * , X,) in X1, X,, * * * , X, with coefficients in GF(q”) such that the sum of powers in each term of f(X) is a multiple of b and the degree of f(X) is pb or less. That is, P(m, e,EL, b) = f(X) j for each term CX;lX;’ . . . X2 of f(X), C is in GF(q’) and 2 vi = jb with 0 5 Jo< p . (4) i=l Since Xf” = Xi, we assume that vi is less than q8 and p is at most equal to mz, where z = (q” - 1)/b. From (3), we have (az”)‘a-l = 1, for 0 < I < b. Therefore, a”’ is an element of GF(q”). It follows from (1) that ai zn+i = aznuii, for 1 5 i 5 m. (5) For 0 5 1 < b, let or denote the set of vectors ez = ((a, i+tn, a, f+zn, a.- , am i+zn) IO 5 i < nl. (6)