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Asymptotically exact bounds on the size of high-order spectral-null codes

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2 Author(s)
G. Freiman ; Sch. of Math., Tel-Aviv Univ., Ramat-Aviv, Israel ; S. Litsyn

The spectral-null code S(n, k) of kth order and length n is the union of n-tuples with ±1 components, having kth-order spectral-null at zero frequency. We determine the exact asymptotic in n behavior of the size of such codes. In particular, we prove that for n satisfying some divisibility conditions, log2|S(n, k)|=n-k 2/2log2n+ck+o(1), where ck is a constant depending only on k and o(1) tends to zero when n grows. This is an improvement on the earlier known bounds due to Roth, Siegel, and Vardy (see ibid., vol40, p.1826-40, 1994)

Published in:

IEEE Transactions on Information Theory  (Volume:45 ,  Issue: 6 )