Codes over rings of size p2 and lattices over imaginary quadratic fields

Abstract

Let ?>0 be a square-free integer congruent to 3 mod 4 and OK the ring of integers of the imaginary quadratic field View the MathML source. Codes C over rings OK/pOK determine lattices ??(C) over K . If p?? then the ring R:=OK/pOK is isomorphic to Fp2 or Fp?Fp. Given a code C over R, theta functions on the corresponding lattices are defined. These theta series ???(C)(q) can be written in terms of the complete weight enumerators of C . We show that for any two ?<?? the first View the MathML source terms of their corresponding theta functions are the same. Moreover, we conjecture that for View the MathML source there is a unique symmetric weight enumerator corresponding to a given theta function. We verify the conjecture for primes p<7, ??59, and small n.