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The joint weight enumerator of an LCD code and its dual. (English) Zbl 1445.94033

A binary code \(C\) is said to be an LCD code if \(C \cap C^\perp = \{\mathbf{0} \}.\) The joint weight enumerator of two codes \(C\) and \(D\) is defined as \(J(C,D) (a,b,c,d) = \sum_{u \in C,v \in D}a^{i(u,v)} b^{j(u,v)} c^{k(u,v)} d^{l(u,v)} \), where \(i(u,v)\) is the number of occurrences of \((0,0)\), \(j(u,v)\) is the number of occurrences of \((0,1)\), \(k(u,v)\) is the number of occurrences of \((1,0)\), and \(l(u,v)\) is the number of occurrences of \((1,1)\).
The authors produce a linear programming bound on the size of an LCD code with a given length and minimum distance by examining the coefficients of the joint weight enumerator using the codes \(C\) and \(C^\perp.\) Additionally, they show how classical invariant theory can be used to show that the joint weight enumerator is an invariant of a matrix group of dimension \(4\) and order \(12\).

MSC:

94B05 Linear codes (general theory)
90C05 Linear programming

Software:

cdd; 4ti2
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Full Text: DOI arXiv

References:

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