Optimal three-weight cyclic codes whose duals are also optimal
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A class of optimal three-weight cyclic codes of dimension 3 over any finite field was presented by Vega [Finite Fields Appl., 42 (2016) 23-38]. Shortly thereafter, Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501-4513] generalized this result by presenting several classes of cyclic codes with either optimal three weights or a few weights. Here we present a new class of optimal three-weight cyclic codes of length q+1 and dimension 3 over any finite field F_q, and show that the nonzero weights are q-1, q, and q+1. We then study the dual codes in this new class, and show that they are also optimal cyclic codes of length q+1, dimension q-2, and minimum Hamming distance 4. Lastly, as an application of the Krawtchouck polynomials, we obtain the weight distribution of the dual codes.
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