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

AbstractLet ℓ>0 be a square-free integer congruent to 3 mod 4 and OK the ring of integers of the imaginary quadratic field K=Q(−ℓ). Codes C over rings OK/pOK determine lattices Λℓ(C) over K. If p∤ℓ 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 θΛℓ(C)(q) can be written in terms of the complete weight enumerators of C. We show that for any two ℓ<ℓ′ the first ℓ+14 terms of their corresponding theta functions are the same. Moreover, we conjecture that for ℓ>p(n+1)(n+2)2 there is a unique symmetric weight enumerator corresponding to a given theta function. We verify the conjecture for primes p<7, ℓ⩽59, and small n