SIAM J. DISCRETE MATH. c  2010 Society for Industrial and Applied Mathematics Vol. 24, No. 3, pp. 964–978 ON REVERSE-FREE CODES AND PERMUTATIONS ∗ ZOLTAN F ¨ UREDI † , IDA KANTOR ‡ , ANGELO MONTI § , AND BLERINA SINAIMERI § Abstract. A set F of ordered k-tuples of distinct elements of an n-set is pairwise reverse free if it does not contain two ordered k-tuples with the same pair of elements in the same pair of coordinates in reverse order. Let F (n, k) be the maximum size of a pairwise reverse-free set. In this paper we focus on the case of 3-tuples and prove lim F (n, 3)/ ( n 3 ) =5/4, more exactly, 5 24 n 3 - 1 2 n 2 - O(n log n) <F (n, 3) ≤ 5 24 n 3 - 1 2 n 2 + 5 8 n, and here equality holds when n is a power of 3. Many problems remain open. Key words. extremal combinatorics, ordered triples, permutations AMS subject classification. 05D05 DOI. 10.1137/090774835 1. The problem. Let k and n be natural numbers, and let X be an n-element underlying set. The set of k-element sequences is denoted by X k , and its cardinality is n k . The set of ordered k-tuples is denoted by X (k) , and the set of k-subsets of X is denoted by ( X k ) . We have X (k) ⊂ X k , |X (k) | = n(n - 1) ... (n - k + 1) = k! ( n k ) , and the cardinality of ( X k ) is ( n k ) (here n ≥ k). Frequently the set X is identified with the set of first n integers [n]= {1,...,n}. A code C is simply a subset of X k ; k is called its length, and |C| is its size. A typical problem in coding theory is finding the maximum size of a code with some local side condition. In this paper we will deal with such a problem, with reverse-free codes. Two sequences x = x 1 ,...,x k and y = y 1 ,...,y k are called reverse free if there are no two coordinates i, j ∈ [k] such that x i = x j but (x i ,x j )=(y j ,y i ). A code F is called pairwise reverse free if any two of its members are reverse free. Let F (n, k) be the maximum cardinality of a pairwise reverse-free code F⊂ [n] k . Let F (n, k) be the maximum cardinality of a pairwise reverse-free code F⊂ [n] (k) ; i.e., when the codewords have no repetition, F is a set of ordered k-tuples from [n]. If one considers the |F | × k matrix M (F ) of F , where the rows are the members of F , then it is reverse free if the rows are distinct and it does not contain a submatrix of type ( ab ba ) with a = b. Many coding theory problems can be formulated in this way; for an extremal problem with forbidden submatrices, see, e.g., [18]. It seems difficult to determine the exact value of F (n, k) and F (n, k), and thus we concentrate on estimating their asymptotic behavior as n tends to ∞ for k fixed. We also solve the first nontrivial cases and determine asymptotically F (n, 3), F (n, 3). Moreover we establish the order of magnitude of F (3,k). ∗ Received by the editors October 23, 2009; accepted for publication (in revised form) June 1, 2010; published electronically August 17, 2010. http://www.siam.org/journals/sidma/24-3/77483.html † R´ enyi Institute of the Hungarian Academy, Budapest, P.O. Box 127, H-1364, Hungary (furedi@ renyi.hu) and the Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 (z-furedi@math.uiuc.edu). This author’s research was supported in part by the Hungarian National Science Foundation under grants OTKA 062321 and 060427 and by the National Science Foundation under grant NFS DMS 06-00303. ‡ Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 (isvejda2@math.uiuc.edu). § Department of Computer Science, Sapienza University of Rome, Via Salaria 113, Rome 00198, Italy (monti@di.uniroma1.it, sinaimeri@di.uniroma1.it). 964