MinT - Best Known (51, 51+38, s)-Nets in Base 16

Best Known (51, 51+38, s)-Nets in Base 16

Digital (51, 89, 526)-net over F₁₆, using

1 times m-reduction [i] based on digital (51, 90, 526)-net over F₁₆, using
- trace code for nets [i] based on digital (6, 45, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (51, 89, 789)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁹, 789, F₁₆, 38) (dual of [789, 700, 39]-code), using
- 138 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 1, 5 times 0, 1, 12 times 0, 1, 25 times 0, 1, 39 times 0, 1, 50 times 0) [i] based on linear OA(16⁸⁰, 642, F₁₆, 38) (dual of [642, 562, 39]-code), using
  - trace code [i] based on linear OA(256⁴⁰, 321, F₂₅₆, 38) (dual of [321, 281, 39]-code), using
    - extended algebraic-geometric code AG_e(F,282P) [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 (51, 89, 230918)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146792 584364 823075 929315 279352 003373 989308 152937 875585 177697 220292 514066 120317 538972 215403 937145 311416 145856 > 16⁸⁹ [i]