Advances in Mathematics of Communications doi:10.3934/amc.2020097 NEW QUANTUM CODES FROM CONSTACYCLIC CODES OVER THE RING R k,m Habibul Islam, Om Prakash ∗ and Ram Krishna Verma Department of Mathematics Indian Institute of Technology Patna Patna- 801 106, India (Communicated by Eimear Byrne) Abstract. For any odd prime p, we study constacyclic codes of length n over the finite commutative non-chain ring R k,m = F p m[u 1 ,u 2 ,...,u k ]/〈u 2 i − 1,u i u j − u j u i 〉 i=j=1,2,...,k , where m, k ≥ 1 are integers. We determine the necessary and sufficient condition for these codes to contain their Euclidean duals. As an application, from the dual containing constacyclic codes, several MDS, new and better quantum codes compare to the best known codes in the literature are obtained. 1. Introduction Quantum computing is a fascinating topic for present research with a higher abil- ity to solve severe problems faster than classical computers. The quantum error- correcting codes are used in the quantum computer to protect the quantum informa- tion from the noises that occurred during communication. After the pioneering work of Shor [35] in 1995, Calderbank et al. [5] proposed a prominent method to obtain quantum error-correcting codes from the classical error-correcting codes. The pri- mary goal of this area is to construct better quantum codes employing state-of-art. In this connection, many significant works have been reported in the literature which provides better quantum codes over the finite fields, see [14, 15, 16, 26, 32]. It is also observed that the linear (e.g., cyclic, constacyclic) codes over finite non-chain rings produced a huge amount of good quantum codes [1, 2, 8, 11, 13, 19, 12, 27, 29, 30]. In 2015, Ashraf and Mohammad [1] studied quantum codes from cyclic codes over F p + vF p . Meantime, Dertli et al. [8] presented some new binary quantum codes obtained from the cyclic codes over F 2 + uF 2 + vF 2 + uvF 2 , and then Ashraf and Mohammad [2] generalized their work over the ring F q + uF q + vF q + uvF q to de- rive new non-binary quantum codes. There are a lot of articles in which good quantum codes are obtained from the cyclic codes on different finite rings, see [11, 13, 19, 23, 31, 33, 32, 34]. On the other side, recently, Gao and Wang [12], Li et al.[27], Ma et al. [29, 30] considered the constacyclic codes over finite non-chain rings and obtained many new and better codes compare to the known codes. Based on the above studies, one can say that the constacyclic codes are a great resource to supply good quantum codes over finite rings. Hence, it is logical to study the 2020 Mathematics Subject Classification: 94B15, 94B05, 94B60. Key words and phrases: Constacyclic code, gray map, self-dual code, quantum code. The research is supported by the University Grants Commission (UGC) and the Council of Scientific & Industrial Research (CSIR), Govt. of India. ∗ Corresponding author: Om Prakash. 1