140 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 1, JANUARY 1998 Bounds on Mixed Binary/Ternary Codes A. E. Brouwer, Heikki O. Ham¨ al¨ ainen,¨ Patric R. J. Osterg¨ ard,˚ Member, IEEE, and N. J. A. Sloane, Fellow, IEEE Abstract— Upper and lower bounds are presented for the , where, for and , maximal possible size of mixed binary/ternary error-correcting we have if and only if and IQ codes. A table up to length is included. The upper bounds . are obtained by applying the linear programming bound to the product of two association schemes. The lower bounds arise from It is trivial to verify that this product scheme indeed is an a number of different constructions. association scheme. The intersection numbers are given by , where , etc., and the dual intersec- Index Terms— Binary codes, clique finding, linear program- ming bound, mixed codes, tabu search, ternary codes. tion numbers by . The adjacency matrices are given by , the idempotents by , and for the eigenmatrix and dual eigenmatrix (defined I. INTRODUCTION by and ) we have ET be the set of all vectors with binary and . Land ternary coordinates (in this order). Let Products of more than two schemes can be defined in an denote Hamming distance on . We study the existence of analogous way (and the multiplication of association schemes large packings in , i.e., we study the function is associative). giving the maximal possible size of a code in with Although product schemes are well known, we cannot find for any two (distinct) codewords . an explicit discussion of their properties or applications. There The dual version of this problem, the existence of small is a short reference in Godsil [15, p. 231] and an only slightly coverings in , has been discussed in [17] and [33]. Both longer one in Dey [11, Sec. 5.10.7]. (We wrote this in 1995. of these problems were originally motivated by the football In the meantime several other applications of product schemes pool problem (see [16]). have come to our attention. See for example [18], [19], [30], We begin by describing the use of product schemes to get [36], and [37].) Another recent paper dealing with mixed codes upper bounds on , and then discuss various con- is [12]. structions and computer searches that provide lower bounds. Our interest in product schemes in the present context stems Among the codes constructed, there are a few (with ) from the fact that the set of mixed binary/ternary vectors with that improve the known lower bounds for ternary codes. binary and ternary coordinate positions does not, in The paper concludes with a table of for general, form an association scheme with respect to Hamming . The first and fourth authors produced a version distance, and so Delsarte’s linear programming bound cannot of this table in 1995 (improving and extending various tables be directly applied there. This was a source of worry to the already in the literature, for example, that in [24]). These fourth author for many years. However, this set does have results were then combined with those of the second and the structure of a product scheme, and so a version of the third authors, who had used computer search and various linear programming bound can be obtained for both designs constructions to obtain lower bounds (many of which were (cf. [37]) and codes. tabulated by the second author already in 1991). The linear programming bound for codes in an arbitrary association scheme can be briefly described as follows. If II. PRODUCTS OF ASSOCIATION SCHEMES (the code we want to study) is a nonempty subset of an association scheme, we can define its inner distribution by Let and be two association schemes, , the average number of codewords with and . at “distance” from a codeword. Clearly, (if is the (For definitions and notation, see [7, ch. 2].) We get a new identity relation), and . A one-line proof1 shows association scheme , the product of these two, by taking that one has (that is, for all ), and thus for the point set, and we obtain the linear programming bound Manuscript received January 10, 1997; revised May 7, 1997. The work of P. R. J. Osterg¨ ard˚ was supported by the Academy of Finland. and A. E. Brouwer is with the Department of Mathematics, Eindhoven Univer- sity of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. The upper bound obtained this way will be referred to as the H. O. Ham¨ al¨ ainen¨ is at Lehtotie 28, 41120 Puuppola, Finland. P. R. J. Osterg¨ ard˚ is with the Department of Computer Science and “pure LP” bound. As we shall see, slightly better results can Engineering, Helsinki University of Technology, P.O. Box 1100, 02015 HUT, 1 Let 1 be the characteristic vector of g. Then, since ij is idempotent Finland. N. J. A. Sloane is with Information Sciences Research Center, AT&T b b P Labs–Research, Florham Park, NJ 07932-0971 USA. jgj@Aj a jgj iij a ij 1 ei1 a 1 ij 1 a jjij 1jj ! HX Publisher Item Identifier S 0018-9448(98)00014-5. i i 0018–9448/98$10.00 1998 IEEE BROUWER et al.: BOUNDS ON MIXED BINARY/TERNARY CODES 141 sometimes be obtained by adding other inequalities that is Indeed, this follows if we again go “to the bottom,” express known to satisfy. everything in terms of , and use the symmetric group on the coordinates. A. The Hamming Scheme Thus the two systems are equivalent over . However, the Of course, the usual Hamming scheme also carries detailed system can be useful i) if it is known that the the structure of a product scheme, for , and it is are integral, e.g., because is linear, or ii) when one can sometimes useful to study nonmixed codes using this product add further constraints, e.g., because one has information on scheme setting, getting separate information on the weights in a residual code. Jaffe [19] has recently obtained a number of the head and in the tail of the codewords, as in the split weight new bounds for binary linear codes by recursive applications enumerator of a code (cf. [29, pp. 149–150]). of this approach. Consider the Hamming scheme as being obtained from the product of and by merging III. COMPARISON WITH EARLIER all relations with into one relation .We RESULTS AND THE CASE have A. Counting In the final section we give tables of upper and lower bounds for codes in the mixed binary/ternary scheme. A table with upper bounds was given in Van Lint Jr. and Van Wee [24]. Pure and linear programming agrees with or improves all the values in their table with four exceptions, namely the parameter sets , where [24] gives while the pure LP bound for any pair with . Indeed, the first holds yields the upper bounds respectively. by definition, the second follows from the third, and the third The upper bound used in these cases in [24] is due to Van follows as soon as we have shown that the right-hand side Wee [43, Theorem 17], and states that if , , and does not depend on the choice of the pair . But that , then , where , follows by viewing all three association schemes involved as and if is even or if is merged versions of powers of : we must show that odd.2 In fact, a stronger result is true. Proposition 3.1: If , and is even or , then . Proof: Let be a code. Count paths for any - vector with . However, since such with , , , where vectors are equivalent under the symmetric group on the is nonzero at a binary coordinate position if is even, and at a coordinates, the right-hand side is independent of , and the ternary coordinate position if is odd. Put if is even, equality follows. and if is odd. For we have choices; given Here we did not need to actually compute the , but since there are choices for ; given and there is at least one in we have choice for . The number of paths is, therefore, at least .On the other hand, there are at most choices for , and given there are at most choices for ,so the number of paths is at most . it follows immediately that in In the four cases mentioned, this yields the bounds respectively. We shall see below (in Proposition 5.10) that the last mentioned bound in fact holds with equality. The “detailed” linear programming bound obtained in the above manner always implies the “ordinary” linear program- ming bound: given any solution of the detailed system B. Linear Programming with Additional Inequalities The preceding results were obtained by studying what hap- for all pens close to the code. In general, one should obtain at least it follows by summing over the pairs with as strong results by adding analogous constraints on the that , where, of course, . with small to the linear program (note that we change Conversely, given any solution of the ordinary system notation here from what is usual in association scheme theory , we find a solution of the detailed system by to what is common in coding theory, and write where the letting previous section had ). 2 Gerhard van Wee has pointed out to us that there is a typographical error for all in the statement of this bound in [43, Theorem 17]. The bound given here (which follows at once from [24, Theorem 9]) is the correct version. 142 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 1, JANUARY 1998 What are the obvious inequalities to add when ? (if ). (The inequalities given earlier for are special Well, no two words of weight can agree in two nonzero cases of those obtained here.) Occasionally also coordinates, so we have a packing problem for triples in a -set, with a prespecified matching of size , where the triples may not cover any edge of the matching.