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We construct trace codes over \(\mathbb {Z}_4\) based on Boolean functions and their support. The Lee weight distribution of these codes is studied by using the Walsh–Hadamard transform of the Boolean functions, and exponential character sums. We obtain few weights codes. In particular, bent and semi-bent functions give three-weight codes.

Trace codes over $${\mathbb {Z}}_4,$$ Z 4 , and Boolean functions