MinT - Best Known (54, 95, s)-Nets in Base 16

Best Known (54, 95, s)-Nets in Base 16

Digital (54, 95, 526)-net over F₁₆, using

1 times m-reduction [i] based on digital (54, 96, 526)-net over F₁₆, using
- trace code for nets [i] based on digital (6, 48, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (54, 95, 782)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁵, 782, F₁₆, 41) (dual of [782, 687, 42]-code), using
- 131 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 1, 5 times 0, 1, 12 times 0, 1, 24 times 0, 1, 37 times 0, 1, 46 times 0) [i] based on linear OA(16⁸⁶, 642, F₁₆, 41) (dual of [642, 556, 42]-code), using
  - trace code [i] based on linear OA(256⁴³, 321, F₂₅₆, 41) (dual of [321, 278, 42]-code), using
    - extended algebraic-geometric code AG_e(F,279P) [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 (54, 95, 252674)-net in base 16, because

1 times m-reduction [i] would yield (54, 94, 252674)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 153916 106604 722771 521650 124026 505875 891121 927457 153581 761169 641271 360810 913731 362084 106163 829270 794009 703091 936576 > 16⁹⁴ [i]