MinT - Best Known (84−51, 84, s)-Nets in Base 16

Best Known (84−51, 84, s)-Nets in Base 16

(84−51, 84, 98)-Net over F₁₆ — Constructive and digital

Digital (33, 84, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 27, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (6, 57, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(84−51, 84, 128)-Net in Base 16 — Constructive

(33, 84, 128)-net in base 16, using

base change [i] based on digital (5, 56, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(84−51, 84, 193)-Net over F₁₆ — Digital

Digital (33, 84, 193)-net over F₁₆, using

net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
  - a curve from the manYPoints database [i]

(84−51, 84, 6734)-Net in Base 16 — Upper bound on s

There is no (33, 84, 6735)-net in base 16, because

1 times m-reduction [i] would yield (33, 83, 6735)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8751 328072 134396 453599 203667 438596 581548 212264 233542 229971 469766 751179 257561 510104 594534 014939 644376 > 16⁸³ [i]