MinT - Best Known (48, 83, s)-Nets in Base 16

Best Known (48, 83, s)-Nets in Base 16

Digital (48, 83, 526)-net over F₁₆, using

1 times m-reduction [i] based on digital (48, 84, 526)-net over F₁₆, using
- trace code for nets [i] based on digital (6, 42, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (48, 83, 803)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸³, 803, F₁₆, 35) (dual of [803, 720, 36]-code), using
- 152 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 1, 5 times 0, 1, 13 times 0, 1, 27 times 0, 1, 44 times 0, 1, 56 times 0) [i] based on linear OA(16⁷⁴, 642, F₁₆, 35) (dual of [642, 568, 36]-code), using
  - trace code [i] based on linear OA(256³⁷, 321, F₂₅₆, 35) (dual of [321, 284, 36]-code), using
    - extended algebraic-geometric code AG_e(F,285P) [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 (48, 83, 307574)-net in base 16, because

1 times m-reduction [i] would yield (48, 82, 307574)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 546 821613 419048 762218 613420 389134 201448 150370 047576 709169 196494 008704 103218 952569 379232 750773 377621 > 16⁸² [i]