Secret Sharing Schemes from Two Families of Cyclic Codes

Abstract

The objectives of this study include proving that ${\widetilde{C}}_{(q,m,\delta_2)}$ and ${\widetilde{C}}_{(q,m,\delta_3)}$, which are Primitive BCH codes, with $m\geq 5$ are minimal codes, and presenting specific examples of secret-sharing schemes based on dual of these codes. To prove that ${\widetilde{C}}_{(q,m,\delta_2)}$ and  $\widetilde{C}_{\left(q,m,\delta_3\right)}$ with $m\geq 5$ are minimal codes, the criterion $\frac{W_{min}}{W_{max}}>\frac{q-1}{q}$ used, where $W_{min}$ and $W_{max}$ are the minimum weight and maximum weight, respectively. Data on the minimum weight and maximum weight of ${\widetilde{C}}_{(q,m,\delta_2)}$ and ${\widetilde{C}}_{(q,m,\delta_3)}$ are obtained from previous research. To give an example of secret-sharing scheme construction based on these codes, the construction method to be used is Massey construction. This research successfully proves that ${\widetilde{C}}_{(q,m,\delta_2)}$ and ${\widetilde{C}}_{(q,m,\delta_3)}$ with $m\geq 5$ are minimal codes. In addition, this research also successfully presents an example of secret-sharing scheme construction based on these codes using Massey's construction.