Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
The Automorphism group of an [18,9,8] quaternary code
Discrete Mathematics - Coding Theory
Introduction to Coding Theory
Self-Dual Codes and Invariant Theory (Algorithms and Computation in Mathematics)
Self-Dual Codes and Invariant Theory (Algorithms and Computation in Mathematics)
The weight distribution of the third-order Reed-Muller code of length 512
IEEE Transactions on Information Theory
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Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ R (where R = 1 + xZ[x]) can be written as f = gn for g ∈ R, n ≥ 2. Let Pn := {gn|g ∈ R} and let µn := n ∏p|nP. We show among other things that (i) for f ∈ R, f ∈ pn, ≡ f (mod µn) ∈ Pn, and (ii) if f ∈ Pn, there is a unique g ∈ Pn with coefficients mod µn/n such that f ≡ gn. In particular, if f ≡ 1 (mod µn) then f ∈ Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E8 in R8) is in Pn if n is of the form 2i3j5k (i ≥ 3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed-Muller code of length 2m is in P2r, (and similarly that the theta series of the Barnes-Wall lattice BW2m is in P2m). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f ∈ Pn (n ≥ 2) with coefficients restricted to the set {1, 2,..... n}.