a. Upon the death of a member or of a former member, there shall be paid to his or her estate, or to such person as he or she has nominated or shall nominate by written designation duly executed and filed with such board during the lifetime of the member:
1. Such member's accumulated deductions, if any; and, in addition thereto;
2. (a) If he or she is a member who is in city-service or is on a civil service preferred eligible list by reason of city-service, unless a pension by payable by the city under the provisions of section 13-149 of this chapter, a sum which, subject to the provisions of paragraph four of subdivision e of section 13-638.4 of this title, shall consist of:
(i) an amount equal to the compensation earnable by such member while a member, during the six months immediately preceding his or her death; or
(ii) if the total number of years in which allowable service was rendered exceeds ten, including service which was allowable during former membership, then an amount equal to the compensation earnable by such member in city-service while a member during the twelve months immediately preceding his or her death; or
(iii) if such member, at the time of his or her death, held a career pension plan position, and if the total number of years in which allowable service was rendered includes twenty or more years of career pension plan qualifying service, including career pension plan qualifying service which was allowable during former membership, then an amount equal to twice the compensation earnable by him or her in city-service while a member during the twelve months immediately preceding his or her death; and
(iv) in addition to the amount payable under item (i), (ii) or (iii) of this subparagraph (a), the reserve-for-increased-take-home-pay, if any.
(b) If the sum of such pension payments made and payable under section 13-149 of this chapter plus the reserve-for-increased-take-home-pay, if any, payable as a lump sum under such section, is a lesser sum, then there shall be paid hereunder the difference between the total of such lesser pension sum and reserve, if any, and the greater amount herein provided as ordinary death benefit.
(c) Where any member, by any designation heretofore or hereafter filed pursuant to the foregoing provisions of this subdivision a and in effect at the time of the death of such member, nominated or shall nominate any person to receive the amount payable under item (i), (ii) or (iii) of subparagraph (a) of this paragraph two, the reserve-for-increased-take-home-pay, if any, of such member payable under item (iv) of such subparagraph (a) shall be paid to the person so nominated.
(d) Payment of the expense of burial not exceeding three hundred dollars to a relative or friend who shall assume responsibility therefor in the absence or failure of the designated beneficiary may be authorized by the comptroller on certificate of the executive director and actuary of the retirement system; such payment by a like amount shall reduce the amount payable to such designated beneficiary or estate.
b. Until the first payment has been made on account of a retirement benefit without optional selection of a member, such member shall be construed by such board to have been in city-service and the benefits provided in this section shall be paid in lieu of the retirement allowance.
c. 1. The member, or on the death of the member, the person nominated by him or her to receive either his or her accumulated deductions, if any, his or her death benefit, or both, may provide by written designation duly executed and filed with such board that the actuarial equivalent of the benefit otherwise payable in a lump sum shall be paid to the person designated in the form of an annuity payable in installments not more often than once a month, the amount of such annuity to be determined at the time of the member's death on the basis of the age of the beneficiary at that time.
2. Where any such designation mentioned in paragraph one of this subdivision c was heretofore or is hereafter filed by a member with respect to a benefit otherwise payable pursuant to the provisions of item (i), (ii) or (iii) of subparagraph (a) of paragraph two of subdivision a of this section, and such designation was or is in effect at the time of the death of such member, or where any such designation was heretofore or is hereafter filed by any person so nominated to receive such benefit, the actuarial equivalent of the reserve-for-increased-take-home-pay, if any, of such member shall be paid to the person named in such designation as beneficiary with respect to such benefit under such item (i), (ii) or (iii), in the form of an annuity and in the manner and in accordance with the method of computation prescribed by paragraph one of this subdivision c.