References
-
W.-H. Choi, Mass formula of self-dual codes over Galois rings GR(
$p^2$ , 2), Korean J. Math. 24 (2016), no. 4, 751-764. https://doi.org/10.11568/kjm.2016.24.4.751 -
W.-H. Choi and Y. H. Park, Self-dual codes over
${\mathbb{Z}}_{p^2}$ of small lengths, Korean J. Math. 25 (2017), no. 3, 379-388. https://doi.org/10.11568/KJM.2017.25.3.379 - S. T. Dougherty, J.-L. Kim, and H. Liu, Constructions of self-dual codes over finite commutative chain rings, Int. J. Inf. Coding Theory 1 (2010), no. 2, 171-190. https://doi.org/10.1504/IJICOT.2010.032133
- S. T. Dougherty and Y. H. Park, Codes over the p-adic integers, Des. Codes Cryptogr. 39 (2006), no. 1, 65-80. https://doi.org/10.1007/s10623-005-2542-x
- F. Q. Gouvea, p-Adic Numbers, second edition, Universitext, Springer-Verlag, Berlin, 1997.
- R. W. Hamming, Error detecting and error correcting codes, Bell System Tech. J. 29 (1950), 147-160. https://doi.org/10.1002/j.1538-7305.1950.tb00463.x
-
A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, The
$Z_4$ -linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994), no. 2, 301-319. https://doi.org/10.1109/18.312154 - S. Han, J.-L. Kim, H. Lee, and Y. Lee, Construction of quasi-cyclic self-dual codes, Finite Fields Appl. 18 (2012), no. 3, 613-633. https://doi.org/10.1016/j.ffa.2011.12.006
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
- J.-L. Kim and Y. Lee, Construction of MDS self-dual codes over Galois rings, Des. Codes Cryptogr. 45 (2007), no. 2, 247-258. https://doi.org/10.1007/s10623-007-9117-y
- J.-L. Kim and Y. Lee, An efficient construction of self-dual codes, Bull. Korean Math. Soc. 52 (2015), no. 3, 915-923. https://doi.org/10.4134/BKMS.2015.52.3.915
- S. Lang, Algebraic Number Theory, second edition, Graduate Texts in Mathematics, 110, Springer-Verlag, New York, 1994.
- R. Lidl and H. Niederreiter, Finite Fields, second edition, Encyclopedia of Mathematics and its Applications, 20, Cambridge University Press, Cambridge, 1997.
- B. R. McDonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974.
- G. H. Norton and A. Salagean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory 46 (2000), no. 3, 1060-1067. https://doi.org/10.1109/18.841186
- G. H. Norton and A. Salagean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Engrg. Comm. Comput. 10 (2000), no. 6, 489-506. https://doi.org/10.1007/PL00012382
- Y. H. Park, The classification of self-dual modular codes, Finite Fields Appl. 17 (2011), no. 5, 442-460. https://doi.org/10.1016/j.ffa.2011.02.010
- V. Pless, The number of isotropic subspaces in a finite geometry, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 39 (1965), 418-421.
- Z.-X. Wan, Finite Fields and Galois Rings, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.