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All our words (strings) are over a fixed alphabet. A square is a subword of the form uu=u 2 , where u is a nonempty word. Two squares are distinct if they are of different shape, not just translates of each other. A word u is primitive if u cannot be written in the form u=v j for some j 2. A square u 2 with u primitive is primitive rooted. Let M(n) denote the maximum number of distinct squares, P(n) the maximum number of distinct primitive rooted squares in a word of length n. We prove: no position in any word can be the beginning of the rightmost occurrence of more than two squares, from which we deduce M(n)<2n for all n>0, and P(n)=n&o(n) for infinitely many n. Academic Press

Abstract. Families of Fibonacci codes and Fibonacci representations are defined. Their main attributes are: (i) robustness, manifesting itself by the local containment of errors; (ii) simple encoding and and decoding. The main applica· tiOD explored is the transmission of binary strings whose length is in aD. unknown range, using robust Fibonacci representations instead of the conventional errorsensitive logarithmic ramp representation. Though the former is asymptotically longer than the latter, the former is actually shorter for very large initial segments of integers.

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