MinT - Best Known (34, 59, s)-Nets in Base 16

Best Known (34, 59, s)-Nets in Base 16

Digital (34, 59, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (34, 60, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 30, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (34, 59, 672)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵⁹, 672, F₁₆, 25) (dual of [672, 613, 26]-code), using
- 25 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 6 times 0, 1, 15 times 0) [i] based on linear OA(16⁵⁴, 642, F₁₆, 25) (dual of [642, 588, 26]-code), using
  - trace code [i] based on linear OA(256²⁷, 321, F₂₅₆, 25) (dual of [321, 294, 26]-code), using
    - extended algebraic-geometric code AG_e(F,295P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

There is no (34, 59, 232902)-net in base 16, because

1 times m-reduction [i] would yield (34, 58, 232902)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6902 044091 117905 932426 559253 346985 805693 886433 006273 502541 621127 352536 > 16⁵⁸ [i]