Projections of Binary Linear Codes onto Larger Fields Jon-Lark Kim * , Keith E. Mellinger and Vera Pless Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA, e-mail: jlkim@math.unl.edu, {kmelling, pless}@math.uic.edu June 12, 2003 Abstract We study certain projections of binary linear codes onto larger fields. These pro- jections include the well-known projection of the extended Golay [24, 12, 8] code onto the Hexacode over GF(4) and the projection of the Reed-Muller code R(2, 5) onto the unique self-dual [8, 4, 4] code over GF(4). We give a characterization of these projec- tions, and we construct several binary linear codes which have best known optimal parameters [20, 11, 5], [40, 22, 8], [48, 21, 12], and [72, 31, 16] for instance. We also relate the automorphism group of a quaternary code to that of the corresponding binary code. Keywords Additive codes, Projection onto larger fields. Mathematics Subject Classification: 94B35 Abbreviated Title: Projections of Binary Linear Codes 1 Introduction The construction of good binary (linear) codes from shorter codes has been widely studied by coding theorists. One of main reasons in this direction is to lower the decoding complexity of the original code. The (u|u + v) construction [17], the projection of Z 4 -linear codes onto non-linear binary codes [14], and the projection of codes over GF(p m ) onto codes over GF(p) are such examples. Each of these constructions applies to a large class of binary codes. * Currently with Dept. of Math. & Stat., Univ. of Nebraska-Lincoln, Lincoln, NE, 68588-0323, USA