The Gray images of $$(1+u)$$ constacyclic codes over $$F_{2^m}[u]/\langle u^{k} \rangle $$
Tóm tắt
Let
$$R_{k}$$
denote the polynomial residue ring
$$F_{2^m}[u]/\langle u^{k} \rangle $$
, where
$$2^{j-1}+1\le k\le 2^{j}$$
for some positive integer
$$j$$
. Motivated by the work in [1], we introduce a new Gray map from
$$R_{k}$$
to
$$F_{2^m}^{2^{j}}$$
. It is proved that the Gray image of a linear
$$(1+u)$$
constacyclic code of an arbitrary length
$$N$$
over
$$R_{k}$$
is a distance invariant linear cyclic code of length
$$2^{j}N$$
over
$$F_{2^m}$$
. Moreover, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over
$$F_{2}$$
and
$$F_{4}$$
are constructed under this Gray map.
Tài liệu tham khảo
Abular, T., Siap, I.: Constacyclic codes over \(F_{2} + uF_{2}\). J. Franklin Inst. 345, 520–529 (2009)
Amarra, M.C.V., Nemenzo, F.R.: On \((1-u)\) cyclic codes over \(F_{p^k}+uF_{p^k}\). Appl. Math. Lett. 21(11), 1129–1133 (2008)
Blake, F.: Codes over integer residue rings. Inf. Control 29(4), 295–300 (1975)
Hammons, A.R., Kumar, P.V., Calderbank, A.R., ASloane, N.J.: The \(Z_{4}\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994)
Kai, X., Zhu, S., Li, P.: \((1+\lambda u)\) Constacyclic codes over \(F_{p}[u]/\langle u^k\rangle \). J. Franklin Inst. 347, 751–762 (2010)
Ling, S., Blackford, T.: \(Z_{p^{k+1}}\)-Linear codes. IEEE Trans. Inf. Theory 48(7), 2592–2605 (2002)
Li, Y., Zhu, S.: A class of constacyclic codes over the ring \(F_{p^m}+uF_{p^m}+\cdots +u^{k-1}F_{p^m}\). J. Hefei Univ. Technol. 35(3), 408–411 (2013). (in chinese)
Nechaev, A.: The Kerdock code in a cyclic form. Discret. Math. Appl. 1(1), 123–139 (1989)
Qian, J.F., Zhang, L.N., Zhu, S.X.: \((1+u)\) Constacyclic and cyclic codes over \(F_{2} + uF_{2}\). Appl. Math. Lett. 19(8), 820–823 (2006)
Qian, J.F.: Cyclic codes over finite rings, wireless communications, networking and mobile computing (WiCOM). In: 7th international conference, 1–4 (2011)
Shankar, P.: On BCH codes over arbitrary integer rings. IEEE Trans. Inf. Theory 25(4), 480–483 (1979)
Sobhani, R., Esmaeili, E.: Some constacyclic and cyclic codes over \(F_{q}[u]/\langle u^{t+1}\rangle \). IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 93(4), 808–813 (2010)
Tapia-Recillas, H., Vega, G.: Some constacyclic codes over \(Z_{2^{k+1}}\) and binary quasi-cyclic codes. Discret. Appl. Math. 128(1), 305–316 (2003)
Udomkavanich, P., Jitman, S.: On the Gray image of \(1+u^{m}\) cyclic codes over \(F_{p^k}+uF_{p^k}+\cdots +u^{m}F_{p^k}\). Int. J. Contemp Math. Sci. 4(26), 1265–1272 (2009)
Wolfmann, J.: Negacyclic and cyclic codes over \(Z_{4}\). IEEE Trans. Inf. Theory 45(7), 2527–2532 (1999)