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.
Some properties such as type, trace description, symmetrized weight enumerator, and generalizedHamming weights of the dual of quaternary Goethals codes are discussed. Furthermore, the shortened codesof quaternary Goet...Some properties such as type, trace description, symmetrized weight enumerator, and generalizedHamming weights of the dual of quaternary Goethals codes are discussed. Furthermore, the shortened codesof quaternary Goethals codes and their dual codes are proved to be cyclic.展开更多
This paper constructs a cyclic Z_4-code with a parity-check matrix similar to that of Goethals code but in length 2~m+ 1, for all m ≥ 4. This code is a subcode of the lifted Zetterberg code for m even. Its minimum Le...This paper constructs a cyclic Z_4-code with a parity-check matrix similar to that of Goethals code but in length 2~m+ 1, for all m ≥ 4. This code is a subcode of the lifted Zetterberg code for m even. Its minimum Lee weight is shown to be at least 10, in general, and exactly 12 in lengths 33, 65. The authors give an algebraic decoding algorithm which corrects five errors in these lengths for m = 5, 6 and four errors for m > 6.展开更多
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.
文摘Some properties such as type, trace description, symmetrized weight enumerator, and generalizedHamming weights of the dual of quaternary Goethals codes are discussed. Furthermore, the shortened codesof quaternary Goethals codes and their dual codes are proved to be cyclic.
文摘This paper constructs a cyclic Z_4-code with a parity-check matrix similar to that of Goethals code but in length 2~m+ 1, for all m ≥ 4. This code is a subcode of the lifted Zetterberg code for m even. Its minimum Lee weight is shown to be at least 10, in general, and exactly 12 in lengths 33, 65. The authors give an algebraic decoding algorithm which corrects five errors in these lengths for m = 5, 6 and four errors for m > 6.
