MinT - Best Known (65−27, 65, s)-Nets in Base 16

Best Known (65−27, 65, s)-Nets in Base 16

Digital (38, 65, 524)-net over F₁₆, using

1 times m-reduction [i] based on digital (38, 66, 524)-net over F₁₆, using
- trace code for nets [i] based on digital (5, 33, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (38, 65, 743)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶⁵, 743, F₁₆, 27) (dual of [743, 678, 28]-code), using
- 94 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 0, 0, 0, 1, 11 times 0, 1, 25 times 0, 1, 49 times 0) [i] based on linear OA(16⁵⁸, 642, F₁₆, 27) (dual of [642, 584, 28]-code), using
  - trace code [i] based on linear OA(256²⁹, 321, F₂₅₆, 27) (dual of [321, 292, 28]-code), using
    - extended algebraic-geometric code AG_e(F,293P) [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 (38, 65, 320097)-net in base 16, because

1 times m-reduction [i] would yield (38, 64, 320097)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 115794 521711 059390 553322 333312 441485 288334 114771 393845 288337 200438 427662 518816 > 16⁶⁴ [i]