Self-dual codes over ${\mathbb Z}_{p^2}$ of small lengths
Main Article Content
Abstract
Self-dual codes of lengths less than 5 over ${\mathbb Z}_p$ are completely classified by the second author [The classification of self-dual modular codes, Finite Fields Appl. 17 (2011), 442-460].The number of such self-dual codes are also determined. In this article we will extend the results to classify self-dual codes over ${\mathbb Z}_{p^2}$ of length less than 5 and give the number of codes in each class. Explicit and complete classifications for small $p$'s are also given.
Article Details
References
[1] J.M.P. Balmaceda, R.A.L. Betty and F.R. Nemenzo, Mass formula for self-dual codes over Zp2 , Discrete Math. 308 (2009), 2984–3002 Google Scholar
[2] K. Betsumiya, S. Georgiou, T.A. Gulliver, M. Harada, and C. Kououvinos, On self-dual codes over some prime fields, Discrete Math. 262 (2009), 37–58. Google Scholar
[3] W. Choi, The classification of self-dual codes over Galois rings of length 4, Ph.D thesis, Kangwon National University, 2017. Google Scholar
[4] J.H. Conway and N.J.A. Sloane, Self-dual codes over the integers modulo 4, J. Comin. Theory Ser. A 62 (1993), 30–45. Google Scholar
[5] R. Lidl and H. Neiderreiter, " Finite fields" in Encyclop. Math. Its Applic. vol. 20. 2nd ed., Cambridge University Press, Cambridge, 1997 Google Scholar
[6] F.J. MacWilliams and N.J.A. Sloane, The theory of error-correcting codes, North-Holland, Amsterdam, 1977. Google Scholar
[7] G. Nebe, E. Rains and N.J.A. Sloane, Self-dual codes and invariant theory, Springer-Verlag, 2006. Google Scholar
[8] Y. H. Park, The classification of self-dual modular codes, Finite Fields Appl. 17 (2011), 442-460 Google Scholar
[9] V.S. Pless, The number of isotropic subspaces in a finite geometry, Atti. Accad. Naz. Lincei Rend. 39 (1965) 418–421. Google Scholar
[10] V.S. Pless, On the uniqueness of the Golay codes, J. Combin. Theory 5 (1968) 215–228. Google Scholar
[11] E. Rains and N.J.A. Sloane, Self-dual codes, in the Handbook of Coding Theory, V.S. Pless and W.C. Huffman, eds., Elsevier, Amsterdam, 1998, 177-294. Google Scholar