The origin of the progenitors of type Ia supernovae (SNe Ia) is still uncertain. The core-degenerate (CD) scenario has been proposed as an alternative way for the production of SNe Ia. In this scenario, SNe Ia are formed at the final stage of common-envelope evolution from a merger of a carbon-oxygen white dwarf (CO WD) with the CO core of an asymptotic giant branch companion. However, the birthrates of SNe Ia from this scenario are still not well determined. In this work, we performed a detailed investigation on the CD scenario based on a binary population synthesis approach. The SN Ia delay times from this scenario are basically in the range of 90Myr-2500Myr, mainly contributing to the observed SNe Ia with short and intermediate delay times although this scenario can also produce some old SNe Ia. Meanwhile, our work indicates that the Galactic birthrates of SNe Ia from this scenario are no more than 20% of total SNe Ia due to more careful treatment of mass transfer. Although the SN Ia birthrates in the present work are lower than those in Ilkov & Soker, the CD scenario cannot be ruled out as a viable mechanism for the formation of SNe Ia. Especially, SNe Ia with circumstellar material from this scenario contribute to 0.7-10% of total SNe Ia, which means that the CD scenario can reproduce the observed birthrates of SNe Ia like PTF 11kx. We also found that SNe Ia happen systemically earlier for a high value of metallicity and their birthrates increase with metallicity.
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