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Geometric approach to the MacWilliams Extension Theorem for codes over modules
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The MacWilliams Extension Theorem states that each linear Hamming isometry of a linear code extends to a monomial map. In this paper an analogue of the extension theorem for linear codes over a module alphabet is observed. A geometric approach to the extendability of isometries is described. For a matrix module alphabet we found the minimum length of a code for which an unextendable Hamming isometry exists. We also proved an extension theorem for MDS codes over a module alphabet.
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Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes
The MacWilliams extension theorem fails for stabilizer codes at every qudit dimension; for qubits the minimal counterexample lives at length 3 and is essentially unique, with minimal full-support counterexamples at le...
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