290 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 1, JANUARY 1998 Here the first inequality is by Part a) of Lemma 4.1. It follows that the number of sequences of type in (the union over all -step strategies ) is at most Since the number of types is at most , we then have But this clearly implies Part 2). Suppose is a deterministic -step strategy, say . Then, implies that (because of the condition in the definition of ). Thus implies , and Here the first inequality is by Part b) of Lemma 4.1. It follows that for any deterministic -step strategy . This, together with the disjointness of the sets , implies Part 3). REFERENCES [1] R. Ahlswede and G. Dueck, “Identification in the presence of feed- back—A discovery of new capacity formulas,” IEEE Trans. Inform. Theory, vol. 35, pp. 30–36, Jan. 1989. [2] , “Identification via channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 15–29, Jan. 1989. [3] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [4] I. Csisz´ ar and J. K¨ orner, Information Theory: Coding Theorems for Discrete Memoryless Systems. New York: Academic, 1981. [5] T. S. Han and S. Verd´ u, “New results in the theory of identification via channels,” IEEE Trans. Inform. Theory, vol. 38, pp. 14–25, Jan. 1992. [6] J. H. B. Kemperman, “Strong converses for a general memoryless channel with feedback,” in Trans. 6th Prague Conf. on Information Theory, Stat. Dec. Fct’s and Rand. Proc., 1973. Goppa Codes and Trace Operator P. V´ eron Abstract—We study Goppa codes, , defined by the polynomial It is shown that the dimension of these codes never reaches the general, well-known, bound for Goppa codes. New bounds are proposed depending on the value of and . Furthermore, we prove that when these codes have only even weights. Index Terms—Goppa codes, parameters of Goppa codes, redundancy of Goppa codes, trace operator. I. INTRODUCTION Binary Goppa codes defined by the polynomial have been introduced in [6]. Their dimension have been studied in [8] and [9], where a new bound has been proposed. In this correspondence, we generalize these results by studying the Goppa Codes which are defined by the polynomial where and are two arbitrary elements of We first show that the usual bound cannot be reached by giving a general new bound. Then, we treat the peculiar cases and Moreoever, we give a general property over the components of the codewords which shows that their weight is even when II. GENERALITIES,DEFINITIONS Definition 2.1: Let be a prime number. Let and be two integers, be a polynomial over , and be a subset of , such that The Goppa code , of length , over , is defined as the set of words such that Let us denote by the degree of , then Proposition 2.2: A parity check matrix of the code is . . . . . . . . . Remark: This matrix satisfies , but its rows are in , so they cannot generate the dual of the code , which is defined over Manuscript received April 8, 1996; revised June 30, 1997. The material in this correspondence was presented in part at the 3rd International Conference on Finite Fields, Glasgow, Scotland, U.K., July 1995. The author is with G.E.C.T., Universit´ e de Toulon et du Var, 83957 La Garde Cedex, France. Publisher Item Identifier S 0018-9448(98)00092-3. 0018–9488/98$10.00  1998 IEEE