Infinite families of MDR cyclic codes over [formula omitted] via constacyclic codes over [formula omitted

Abstract

We study α-constacyclic codes over the Frobenius non-chain ring R≔Z4[u]∕〈u2−1〉 for any unit α of R. We obtain new MDR cyclic codes over Z4 using a close connection between α-constacyclic codes over R and cyclic codes over Z4. We first explicitly determine generators of all α-constacyclic codes over R of odd length n for any unit α of R. We then explicitly obtain generators of cyclic codes over Z4 of length 2n by using a Gray map associated with the unit α. This leads to a construction of infinite families of MDR cyclic codes over Z4, where a MDR code means a maximum distance with respect to rank code in terms of the Hamming weight or the Lee weight. We obtain 202 new cyclic codes over Z4 of lengths 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50 and 54 by implementing our results in Magma software; some of them are also MDR codes with respect to the Hamming weight or the Lee weight.