Abstract
In this study the definition and properties of finite chain ring $F_2+υF_2$ are given, as well as its complete weight enumerator and symmetric weight enumerator. And on this basis by introducing a special variable <em>t</em> as a new variable method, to study the linear codes over finite chain rings with dual codes on more than two kinds of weight enumerators related identities.
Highlights
A great deal of attention has been paid to codes over finite rings from the 1990s since a landmark paper (Hammons et al, 1994), which showed that certain nonlinear binary codes can be constructed from Z4-linear codes via the Gray map and that nonlinear binary codes (Preparata and Kerdock codes) satisfy with MacWilliams identity
Define 1: Suppose C is a linear code of length n over R, where r is one element of R
In order to obtain two weight enumerators of MacWilliams identity, we introduce a special variable t
Summary
A great deal of attention has been paid to codes over finite rings from the 1990s since a landmark paper (Hammons et al, 1994), which showed that certain nonlinear binary codes can be constructed from Z4-linear codes via the Gray map and that nonlinear binary codes (Preparata and Kerdock codes) satisfy with MacWilliams identity. Yildiz and Karadeniz (2010) made a research on the linear codes and the MacWilliams identity of the complete weight enumerator over the ring F2+ uF2+νF2+uνF2. Define 1: Suppose C is a linear code of length n over R, where r is one element of R. X wr (c) r c∈C r∈R as the complete weight enumerator of the linear code C. In order to introduce the concept of the symmetric weight enumerator, the elements of ring R should be classified. : SweC ( X 0 , X1, X 2 ) = CweC ( X I (0) , X I (1) , X I (1+v) , X I (v) ) can be called as the symmetric weight enumerator of code C
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