MinT - Best Known (9, 9+53, s)-Nets in Base 128

Best Known (9, 9+53, s)-Nets in Base 128

(9, 9+53, 288)-Net over F₁₂₈ — Constructive and digital

Digital (9, 62, 288)-net over F₁₂₈, using

net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
  - a function field by SÃ©mirat [i]

(9, 9+53, 289)-Net in Base 128

(9, 62, 289)-net in base 128, using

2 times m-reduction [i] based on (9, 64, 289)-net in base 128, using
- base change [i] based on digital (1, 56, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(9, 9+53, 7287)-Net in Base 128 — Upper bound on s

There is no (9, 62, 7288)-net in base 128, because

1 times m-reduction [i] would yield (9, 61, 7288)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 347 340634 906118 206885 779504 979914 247260 927964 736668 681252 448010 482084 318469 328866 820111 111721 075122 942583 613277 387562 410777 278940 > 128⁶¹ [i]