A characteristic property of self-orthogonal codes and its application to lattices
Zhe-Xian Wan · Bulletin of the Belgian Mathematical Society - Simon Stevin · 1998
Let p be an odd prime, ζ = e 2πi/p , D be the ring of algebraic integers in the field Q(ζ), and P = (1ζ) be the principal ideal of D generated by 1ζ.For a p-ary linear code C of length n, define the lattice Λ C = {p -1/2 (c + z) | c ∈ C, z ∈ P n }.It is proved that Λ C is even if and only if C is self-orthogonal and that Λ C is even unimodular if and only if C is self-dual.The proof rests on the following remark that for an odd prime power q a q-ary linear code C is self-orthogonal if and only if c • c = 0 for all c ∈ C. Finally, irreducible root lattices arising as Λ C from p-ary linear codes C are completely determined.