In this work, doubly extended linearized Reed–Solomon codes and triply extended Reed–Solomon codes are generalized. We obtain a general result in which we characterize when a multiply extended code for a general metric attains the Singleton bound. We then use this result to obtain several families of doubly extended and triply extended maximum sum-rank distance (MSRD) codes that include doubly extended linearized Reed–Solomon codes and triply extended Reed–Solomon codes as particular cases. To conclude, we discuss when these codes are one-weight codes.
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