TY - JOUR
T1 - Infinite families of MDR cyclic codes over Z4 via constacyclic codes over Z4[u]∕〈u2−1〉
AU - Han, Nayoung
AU - Kim, Bohyun
AU - Kim, Boran
AU - Lee, Yoonjin
N1 - Publisher Copyright:
© 2019 Elsevier B.V.
PY - 2020/3
Y1 - 2020/3
N2 - 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.
AB - 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.
KW - Constacyclic code
KW - Cyclic code
KW - Frobenius non-chain ring
KW - Gray map
KW - MDR code
UR - http://www.scopus.com/inward/record.url?scp=85076011838&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2019.111771
DO - 10.1016/j.disc.2019.111771
M3 - Article
AN - SCOPUS:85076011838
SN - 0012-365X
VL - 343
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - 3
M1 - 111771
ER -