416 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 1, JANUARY 2001 Since by bound (2), it follows that , , and . It follows from using (3) that Using bound (4) and, therefore, and thus and and, therefore, . ACKNOWLEDGMENT The author wishes to thank Prof. M. Bossert for his hospitality. The author also wishes to thank the anonymous referee for careful reading of the manuscript and for helpful comments and suggestions. This work was done during the visit of the author at Ulm University, Germany. REFERENCES [1] G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein, Covering Codes. Amsterdam, The Netherlands: North-Holland, Elsevier Science B.V., 1997. [2] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [3] E. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 1968. [4] F. R. Kschischang and S. Pasupathy, “Some ternary and quaternary codes and associated sphere packings,” IEEE Trans. Inform. Theory, vol. 38, pp. 227–246, Mar. 1992. [5] E. Velikova, “Bounds on the covering radius of linear codes,” C. R. l’Acad. Bulg. des Sci., vol. 41, pp. 13–16, 1988. [6] H. Mattson Jr., “An improved upper bound on covering radius,” in Lec- ture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 1986, pp. 90–106. [7] T. Baicheva and E. Velikova, “Covering radii of ternary linear codes of small dimensions and codimensions,” IEEE Trans. Inform. Theory, vol. 43, pp. 2057–2061, Nov. 1997. Linear Codes with Covering Radius and Codimension Alexander A. Davydov and Patric R. J. Östergård, Member, IEEE Abstract—Let denote a linear code over with length , codimension , and covering radius . We use a modification of con- structions of and codes to produce infinite families of good codes with covering radius and and codimension . Index Terms—Bounds on codes, covering code, lengthening, linear code, projective geometry. I. INTRODUCTION We denote the finite field of size by , where is a prime power, and vectors of length with elements from by . Moreover, . A linear code in with dimension (so the codimension is ) and covering radius is said to be an code. If a code has covering radius , then all words in can be obtained as a linear combination of at most columns of its parity check matrix. The minimum length such that an code exists is denoted by . For a survey of covering codes, see [3]. If and all words in can be obtained as a linear combina- tion with nonzero coefficients of at least columns of the parity check matrix of an code, we say that it is an code. If all words except the all-zero word can be obtained in this way, we say that it is an code. Respective partitions of the columns such that a required linear combination can always be obtained with the columns belonging to different subsets are called -parti- tions and -partitions. Earlier work on linear covering codes has mainly concerned binary codes and -ary codes with covering radius ; see [3, Chs. 5 and 7]. Ternary codes with covering radius are considered in [1], [4], [9], and short codes with covering radius over various fields are considered in [7]. In this work, codes over arbitrary fields with and are studied. Using a modification of earlier constructions of and codes, lengthening constructions are applied to obtain infinite families of good codes of codimension . Some special properties of the starting codes make these lengthening constructions effective. Some matrices with special properties are studied in Section II. The matrices are used as building blocks to construct and codes in Section III. These codes have the same main parameters as codes constructed earlier in [2], [4], [7], but they also have some interesting partitioning properties. These properties are necessary in applying the lengthening constructions that are discussed in Sections IV and V. Infinite families of codes improving on the results in the literature are then obtained. An updated table of is given for . Manuscript received July 6, 1999; revised November 11, 1999. This work was supported by the Academy of Finland under Grant 44517. A. A. Davydov is with the Institute for Information Transmission Problems, Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 101447, Russia (e-mail: adav@iitp.ru). P. R. J. Östergård is with the Department of Computer Science and Engi- neering, Helsinki University of Technology, P.O. Box 5400, 02015 HUT, Fin- land (e-mail: patric.ostergard@hut.fi). Communicated by P. Solé, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(01)00597-1. 0018–9448/01$10.00 © 2001 IEEE