We explain the algorithms that we have implemented to show that all integers congruent to $4$ modulo $80$ in the interval $[6\times {10}^{12};2.17\times {10}^{14}]$ are sums of five fourth powers, and that all integers congruent to $6,21$ or $36$ modulo $80$ in the interval $[6\times {10}^{12};1.36\times {10}^{23}]$ are sums of seven fourth powers. We also give some results related to small sums of biquadrates. Combining with the Dickson ascent method, we deduce that all integers in the interval $[13793;{10}^{245}]$ are sums of $16$ biquadrates.