Upon the death before retirement of a member while in service or while physically or mentally incapacitated for the performance of his duty, if such incapacity has been continuous from discontinuance of service, or within one month after discontinuance of service unless the member’s accumulated contributions have been paid to the member pursuant to Section 31628, the retirement system is liable for a death benefit which shall be paid:
(a) As provided in Section 31765 or 31765.1 or 31765.11 or 31781.1 or 31787, if the surviving spouse or guardian of one or more of the surviving children of the member so elects, or
(b) If no election is made pursuant to Section 31765, or 31765.1, or 31765.11, or 31781.1, or 31787 and no person is entitled to a survivor’s allowance pursuant to Article 15.5 (commencing with Section 31841) or Article 16 (commencing with Section 31861) to such person as he nominates by written designation duly executed and filed with the board, before the death of the member, or
(c) If no such election is made, and a parent as defined in Article 15.5 but no other person is entitled to a survivor’s allowance pursuant to Article 15.5 or 16 and a surviving spouse or child is designated as beneficiary, to such surviving spouse or child, or
(d) If no such election is made, no person is entitled to a survivor’s allowance pursuant to Article 15.5 or 16, and a member has not nominated a beneficiary, to his estate.