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

  • Authors:

  • T. Shaska;C. Shor;S. Wijesiri

  • Affiliations:

  • Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, United States and Department of Computer Science and Engineering, University of Vlora, Albania;Department of Mathematics, Western New England College, 1215 Wilbraham Road, Springfield, MA 01119, United States;Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, United States

  • Venue:
  • Finite Fields and Their Applications
  • Year:
  • 2010

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Abstract

Let @?0 be a square-free integer congruent to 3 mod 4 and O"K the ring of integers of the imaginary quadratic field K=Q(-@?). Codes C over rings O"K/pO"K determine lattices @L"@?(C) over K. If p@?@? then the ring R:=O"K/pO"K is isomorphic to F"p"^"2 or F"pxF"p. Given a code C over R, theta functions on the corresponding lattices are defined. These theta series @q"@L"""@?"("C")(q) can be written in terms of the complete weight enumerators of C. We show that for any two @?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