This article extended the properties of the idempotent generator of cyclic codes to polycyclic codes over the finite field $ \mathbb{F}_q $. In addition, the check matrix of polycyclic codes was provided over $ \mathbb{F}_q $. Specifically, it has been proven that the constacyclic code is an $ {{{\rm{MDS}}}} $ code over $ \mathbb{F}_q $ if and only if its annihilator dual code is also an $ {{{\rm{MDS}}}} $ code. Finally, we have provided some examples of good codes.
Citation: Wei Qi. The polycyclic codes over the finite field $ \mathbb{F}_q $[J]. AIMS Mathematics, 2024, 9(11): 29707-29717. doi: 10.3934/math.20241439
This article extended the properties of the idempotent generator of cyclic codes to polycyclic codes over the finite field $ \mathbb{F}_q $. In addition, the check matrix of polycyclic codes was provided over $ \mathbb{F}_q $. Specifically, it has been proven that the constacyclic code is an $ {{{\rm{MDS}}}} $ code over $ \mathbb{F}_q $ if and only if its annihilator dual code is also an $ {{{\rm{MDS}}}} $ code. Finally, we have provided some examples of good codes.
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Wei Qi. The polycyclic codes over the finite field $ \mathbb{F}_q $[J]. AIMS Mathematics, 2024, 9(11): 29707-29717. doi: 10.3934/math.20241439
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