Superconducting properties in HgBa{sub 2}Ca{sub 2}Cu{sub 3}O{sub 8+{delta}}, with a transition temperature of {ital T}{sub {ital c}}=133 K, have been investigated by measurements of the nuclear spin-lattice relaxation rate, {sup 63}(1/{ital T}{sub 1}). It has been found that the {ital T}{sub 1}{ital T}=const behavior, observed well below {ital T}{sub {ital c}}, arises from combined relaxation channels to vortex cores, as well as from a residual density of states (DOS), {ital N}{sub res}, at the Fermi level associated with the gapless superconductivity. From the value of {ital T}{sub 1}{ital T}=const in the latter channel, the DOS fraction normalized by its value at {ital T}{sub {ital c}},{ital N}{sub res}/{ital N}{sub {ital T}{sub {ital c}}}, has been deduced to be as small as {approximately}0.05. Based on the two-dimensional gapless {ital d}-wave model, the {ital T}{sub {ital c}} reduction rate, {Delta}{ital T}{sub {ital c}}/{ital T}{sub {ital c}0}, is estimated to be as small as 0.02 for {ital N}{sub res}/{ital N}{sub 0}{approximately}0.05. It is pointed out that these measures of the quality of HaBa{sub 2}Ca{sub 2}Cu{sub 3}O{sub 8+{delta}} explain why {ital T}{sub {ital c}} reaches such a high value. {copyright} {ital 1996 The American Physical Society.}