Hulls of <inline-formula><tex-math id="M1">$\mathbb Z_p\mathbb Z_p[v] $</tex-math></inline-formula>-cyclic codes and construction of EAQECCs

In this paper, we study the hulls of $ \mathbb{Z}_p\mathbb{Z}_p[v] $- cyclic codes for any prime $ p $, where $ v^2 = v $. Firstly, we derive the generator polynomials of the hulls of separable cyclic codes (SCCs) over $ \mathbb{Z}_p\mathbb{Z}_p[v] $ and determine their dimensions using the concept of a generating function in combinatorics. We also enumerate the SCCs over $ \mathbb{Z}_p\mathbb{Z}_p[v] $ with hulls of a fixed dimension. Next, we present the generator polynomials of the hulls of non-separable cyclic codes (NSCCs) over $ \mathbb{Z}_2\mathbb{Z}_2[v] $ with coprime odd block lengths and determine their dimensions. Additionally, we enumerate the NSCCs over $ \mathbb{Z}_2\mathbb{Z}_2[v] $ with hulls of a fixed dimension. As an application, we use the hulls of NSCCs over $ \mathbb{Z}_2\mathbb{Z}_2[v] $ to construct some good entanglement-assisted quantum error-correcting codes (EAQECCs).