LAUSR

Mutually unbiased maximally entangled bases (MUMEBs) in bipartite systems have a close relation with unitary 2-design and have attracted much attention in recent years. In the paper, we construct MUMEBs in $$\mathbb {C}^q\otimes \mathbb {C}^q$$Cq⊗Cq with q a power of an odd prime number. For this purpose, we introduce the notation of trace-2 excluded subset of the special linear group $$SL(2,\mathbb {F}_q)$$SL(2,Fq) over the finite field $$\mathbb {F}_q$$Fq and establish a relation between a trace-2 excluded subset and a set of MUMEBs in $$\mathbb {C}^q\otimes \mathbb {C}^q$$Cq⊗Cq. Under this relation, we prove that $$M(q,q)\ge \dfrac{q^2-1}{2}$$M(q,q)≥q2-12 by constructing trace-2 excluded subsets in $$SL(2,\mathbb {F}_q)$$SL(2,Fq), which highly raises the lower bound of M(q, q) given in Liu et al. (Quantum Inf Process 16(6):159, 2017) and Xu (Quantum Inf. Process 16(3):65, 2017).