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Abstract. In 1963, O'Meara solved the classification problem for lattices over dyadic local fields in terms of Jordan decompositions. In this paper we translate his result in terms of good BONGs. BONGs (bases of norm generators) were introduced in 2003 as a new way of describing lattices over dyadic local fields. This result and the notions we introduce here are a first step towards a solution of the more difficult problem of representations of lattices over dyadic fields.

For each positive integer n, let g Z (n) be the smallest integer such that if an integral quadratic form in n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z (n) squares of integral linear forms. We show that as n goes to infinity, the growth of g Z (n) is at most an exponential of √ n. Our result improves the best known upper bound on g Z (n) which is in the order of an exponential of n. We also define an analogous number g * O (n) for writing hermitian forms over the ring of integers O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1, we show that the growth of g * O (n) is at most an exponential of √ n. We also improve results of Conway-Sloane [2] and Kim-Oh [14] on s-integral lattices.

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Abstract. In this paper we give necessary and sufficient conditions for the representations of quadratic lattices over arbitrary dyadic fields. Our result is given in terms of Bases of Norm Generators (BONGs, for short). However, they can be translated in terms of the more traditional Jordan decompositions.