Abstract:
In the paper, we have considered the application of the Plotkin construction to codes associated with function fields of algebraic curves over finite fields. The main focus has been on analyzing the parameters of the new code $\mathcal{C}$, which was obtained by combining two algebraic geometry codes $\mathcal{C}_1$ and $\mathcal{C}_2$. An explicit construction of both the generator and parity-check matrices is presented, and formulas for the minimum distance and dimension of the resulting code are derived. Moreover, we have discussed the construction of an error-correcting pair $(\mathcal{A}, \mathcal{B})$ for the code $\mathcal{C}$. This allows the use of efficient algebraic decoding methods when working with this class of codes. In particular, if error-correcting pairs exist for the original codes $\mathcal{C}_1$ and $\mathcal{C}_2$, then a corresponding pair can be explicitly constructed for the combined code $\mathcal{C}$, preserving the geometric structure and enabling polynomial-time decoding algorithms. This approach opens up possibilities for building longer codes with controlled parameters, while maintaining good error-correcting capabilities and allowing for practical implementation in coding theory and cryptography.
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