Abstract
In [4] it was shown, that the weight enumerators of two binary ℤ4-linearly dual codes satisfy the McWilliams identity (i.e. these codes are formally dual). If we consider an arbitrary Galois ring R=GR(q 2, p 2) of characteristic p 2 and a pair of R-linearly dual codes over a Galois field GF(q) this result is not preserved. We propose the approach to correcting this disadvantage. The titled codes are presented as codes inthe alphabet ℜ=RS q (q,2), being a Reed-Solomon code. The appropriate exact weight enumerators of these codes are reduced to some projective weight enumerators (obtained by identifying of variables) which satisfy the McWilliams identity for linear codes over GF(q). We discuss ways of “optimal” identifying of variables such that the corresponding projective weight enumerators allow us to construct complete weight enumerators of the initial codes over GF(q).
This work was supported in part by grant RFFR 96-0100627.
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Nechaev, A.A., Kuzmin, A.S. (1997). Formal duality of linearly presentable codes over a Galois field. In: Mora, T., Mattson, H. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1997. Lecture Notes in Computer Science, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63163-1_21
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DOI: https://doi.org/10.1007/3-540-63163-1_21
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