In this paper, we study λ-constacyclic codes over the ring R = Z4 ＋ uZ4, where u^2 = 0, for λ= 1 ＋ 3u and 3 ＋ u. We introduce two new Gray maps from R to Z4^4 and show that the Gray images of λ-constacyclic cod...In this paper, we study λ-constacyclic codes over the ring R = Z4 ＋ uZ4, where u^2 = 0, for λ= 1 ＋ 3u and 3 ＋ u. We introduce two new Gray maps from R to Z4^4 and show that the Gray images of λ-constacyclic codes over R are quasi-cyclic over Z4. Moreover, we present many examples of λ-constacyclic codes over R whose Z4-images have better parameters than the currently best-known linear codes over Z4.展开更多
The characteristic of Quaternary codes is analyzed. The rule of distinguishing triangle direction is given out. An algorithm of neighbor finding by decomposing the Quaternary code from back to front is presented in th...The characteristic of Quaternary codes is analyzed. The rule of distinguishing triangle direction is given out. An algorithm of neighbor finding by decomposing the Quaternary code from back to front is presented in this paper. The contrastive analysis of time complexity between this algorithm and Bartholdi＇s algorithm is approached. The result illustrates that the average consumed time of this algorithm is about 23.66% of Bartholdi＇s algorithm.展开更多
The 2-adic representations of codewords of the dual of quaternary Goethals code are given. By the 2-adic representations, the binary image of the dual of quaternary Goethals code under the Gray map is proved to be the...The 2-adic representations of codewords of the dual of quaternary Goethals code are given. By the 2-adic representations, the binary image of the dual of quaternary Goethals code under the Gray map is proved to be the nonlinear code constructed by Goethals in 1976.展开更多
The trace representation of the dual of quaternary Goethals code G (m) is given .It is proved that the shortened code of G (m) is cyclic and its generators are shown.
Constant weight codes (CWCs) are an important class of codes in coding theory. Generalized Steiner systems GS(2, k, v, g) were first introduced by Etzion and used to construct optimal nonlinear CWCs over an alphabet o...Constant weight codes (CWCs) are an important class of codes in coding theory. Generalized Steiner systems GS(2, k, v, g) were first introduced by Etzion and used to construct optimal nonlinear CWCs over an alphabet of size g+1 with minimum Hamming distance 2k - 3, in which each codeword has length v and weight k. In this paper, Weil's theorem on character sum estimates is used to show that there exists a GS(2,4, v, 3) for any prime v≡1 (mod 4) and v > 13. From the coding theory point of view, an optimal nonlinear quaternary (v, 5,4) CWC exists for such a prime v.展开更多
In this paper, the quaternary Delsarte-Goethals code D(m,δ) and its dual code G(m,δ) are discussed. The type and the trace representation are given for D(m,δ), while the type and the minimum Lee weight are determin...In this paper, the quaternary Delsarte-Goethals code D(m,δ) and its dual code G(m,δ) are discussed. The type and the trace representation are given for D(m,δ), while the type and the minimum Lee weight are determined for G(m,δ). The shortened codes of D(m,δ) and G(m,δ) are proved to be 4-cyclic. The binary image of D(m,δ) is proved to be the binary Delsarte-Goethals code DG(m+1,δ),and the essential difference between the binary image of G(m,δ) and the binary Goethals-Delsarte codeGD(m+1,δ) is exhibited. Finally, the decoding algorithms of D■(m,δ) and G(m,δ) are discussed.展开更多
文摘In this paper, we study λ-constacyclic codes over the ring R = Z4 ＋ uZ4, where u^2 = 0, for λ= 1 ＋ 3u and 3 ＋ u. We introduce two new Gray maps from R to Z4^4 and show that the Gray images of λ-constacyclic codes over R are quasi-cyclic over Z4. Moreover, we present many examples of λ-constacyclic codes over R whose Z4-images have better parameters than the currently best-known linear codes over Z4.
基金Supported by the Natural Science Foundation of China （No. 40771169 No.40471108 No.40701152）.
文摘The characteristic of Quaternary codes is analyzed. The rule of distinguishing triangle direction is given out. An algorithm of neighbor finding by decomposing the Quaternary code from back to front is presented in this paper. The contrastive analysis of time complexity between this algorithm and Bartholdi＇s algorithm is approached. The result illustrates that the average consumed time of this algorithm is about 23.66% of Bartholdi＇s algorithm.
文摘The 2-adic representations of codewords of the dual of quaternary Goethals code are given. By the 2-adic representations, the binary image of the dual of quaternary Goethals code under the Gray map is proved to be the nonlinear code constructed by Goethals in 1976.
文摘The trace representation of the dual of quaternary Goethals code G (m) is given .It is proved that the shortened code of G (m) is cyclic and its generators are shown.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10471127)for the first authorby Tianyuan Mathematics Foundation of NSFC(Grant No.A0324644)Guangxi Science Foundation and the Foundation of the Education Department of Guangxi Province for the second author.
文摘Constant weight codes (CWCs) are an important class of codes in coding theory. Generalized Steiner systems GS(2, k, v, g) were first introduced by Etzion and used to construct optimal nonlinear CWCs over an alphabet of size g+1 with minimum Hamming distance 2k - 3, in which each codeword has length v and weight k. In this paper, Weil's theorem on character sum estimates is used to show that there exists a GS(2,4, v, 3) for any prime v≡1 (mod 4) and v > 13. From the coding theory point of view, an optimal nonlinear quaternary (v, 5,4) CWC exists for such a prime v.
文摘In this paper, the quaternary Delsarte-Goethals code D(m,δ) and its dual code G(m,δ) are discussed. The type and the trace representation are given for D(m,δ), while the type and the minimum Lee weight are determined for G(m,δ). The shortened codes of D(m,δ) and G(m,δ) are proved to be 4-cyclic. The binary image of D(m,δ) is proved to be the binary Delsarte-Goethals code DG(m+1,δ),and the essential difference between the binary image of G(m,δ) and the binary Goethals-Delsarte codeGD(m+1,δ) is exhibited. Finally, the decoding algorithms of D■(m,δ) and G(m,δ) are discussed.
