Abstract
This work considers relations, which connect weight enumerator of a linear code \( \mathcal{L} \subset \mathbb{F}_p^n \) and weight enumerator of code \( \mathcal{K} \) , lying between \( \mathcal{L} \) and \( \mathbb{F}_p^n :\mathcal{L} \subseteq \mathcal{K} \subseteq \mathbb{F}_p^n \) . The connection is establish by code \( \mathcal{L}^ \bot \left( \mathcal{K} \right) \) , which is dual code of code \( \mathcal{L} \) in the code \( \mathcal{K} \) . It has been show, that there is a code \( \mathcal{L}^ \bot \left( \mathcal{K} \right) \) , which dimension is k−l, where l and k are dimensions of \( \mathcal{L} \) and \( \mathcal{K} \) respectively. These relations for weight enumerators generalise the MacWilliams relation, in which \( \mathcal{K} = \mathbb{F}_p^n \).
The new relations are written for linear code of length 2n of code \( \mathcal{L} = \left\{ {\left( {a\left| {a + b} \right.} \right),a \in \mathcal{L}_1 ,b \in \mathcal{L}_2 } \right\} \) , where \( \mathcal{L}_1 ,\mathcal{L}_2 \subseteq \mathbb{F}_p^n \) for two different comprehensive codes \( \mathcal{K} \). As a result we obtain two different expressions for weight enumerator of code \( \mathcal{L} \).
As example we consider a case \( \mathcal{L}_1 = RM_{m - 2.m} ,\mathcal{L}_2 = RM_{1.m} \), where RM r.m is the r-th order Reed-Muller code of lenght n = 2m.1
Thise research was support by The Royal Swedesh akademy of Sciences
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References
F.W. MacWilliams and N.W.A. Sloane, The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).
J. Simonis, MacWillams Identities and Coordinate Partitions, Linear Algebra and its Applications, Vol. 216 (1995), pp. 81–91.
G. David Forney, Jr. Transforms and Groups, Codes, curves, and signals (Urbana, IL, 1997), 79–97, Kluwer Acad. Publ., Boston. MA. 1998.
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Kazarin, L.S., Sidelnikov, V.M., Gashkov, I.B. (2003). Relative Duality in MacWilliams Identity. In: Fossorier, M., Høholdt, T., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2003. Lecture Notes in Computer Science, vol 2643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44828-4_13
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DOI: https://doi.org/10.1007/3-540-44828-4_13
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