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

Tanush Shaska, Caleb Shor, S. Wijesiri

I-AIMS paper 2007-2 ยท Published in Finite Fields Appl. 16 (2010), 75โ€“87 ยท 2010

Let โ„“ > 0 be a square-free integer congruent to 3 mod 4 and ๐’ช_K the ring of integers of the imaginary quadratic field K = โ„š(โˆš(โˆ’โ„“)). Codes C over rings ๐’ช_K/p๐’ช_K determine lattices ฮ›_โ„“(C) over K. If p โˆค โ„“ then the ring R:= ๐’ช_K/p๐’ช_K is isomorphic to ๐”ฝ_(pยฒ) or ๐”ฝโ‚š ร— ๐”ฝโ‚š. Given a code C over R, theta functions on the corresponding lattices are defined. These theta series ฮธ_(ฮ›_โ„“(C)) can be written in terms of the complete weight enumerator of C. We show that for any two โ„“ < โ„“โ€ฒ the first (โ„“+1)/4 terms of their corresponding theta functions are the same. Moreover, we conjecture that for โ„“ > (p(n+1)(n+2))/2 there is a unique complete weight enumerator corresponding to a given theta function. We verify the conjecture for primes p < 7 and โ„“ โ‰ค 59.

MSC classes: Primary 94B40; Secondary 11F27, 94B05, 11T71

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References

  1. C. Bachoc, Applications of coding theory to the construction of modular lattices. J. Combin. Theory Ser. A 78 (1997), no. 1, 92โ€“119.
  2. K. S. Chua, Codes over GF(4) and F2ร—F2 and Hermitian lattices over imaginary quadratic fields. Proc. Amer. Math. Soc. 133 (2005), no. 3, 661โ€“670 (electronic).
  3. J. Leech and N. J. A. Sloane, Sphere packing and error-correcting codes, Canadian J. Math., 23, (1971), 718-745.
  4. F. J. MacWilliams, N. J. A. Sloane, The theory of error-correcting codes. II. North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. pp. iโ€“ix and 370โ€“762.
  5. F. J. MacWilliams, N. J. A. Sloane, The theory of error-correcting codes. I. North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. pp. iโ€“xv and 1โ€“369.
  6. T. Shaska and C. Shor, Codes over Fp2 and Fpร—Fp, lattices, and corresponding theta functions. Advances in Coding Theory and Cryptology, vol 3. (2007), pg. 70-80.
  7. T. Shaska and S. Wijesiri, Codes over rings of size four, Hermitian lattices, and corresponding theta functions, Proc. Amer. Math. Soc., 136 (2008), 849-960.

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