MinT - Best Known (84−48, 84, s)-Nets in Base 16

Best Known (84−48, 84, s)-Nets in Base 16

(84−48, 84, 130)-Net over F₁₆ — Constructive and digital

Digital (36, 84, 130)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 30, 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]
2. digital (6, 54, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(84−48, 84, 177)-Net in Base 16 — Constructive

(36, 84, 177)-net in base 16, using

3 times m-reduction [i] based on (36, 87, 177)-net in base 16, using
- base change [i] based on digital (7, 58, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(84−48, 84, 193)-Net over F₁₆ — Digital

Digital (36, 84, 193)-net over F₁₆, using

t-expansion [i] based on 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]

(84−48, 84, 10694)-Net in Base 16 — Upper bound on s

There is no (36, 84, 10695)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 140145 328402 933385 699127 818795 013594 316394 963410 726966 907367 305212 981685 086944 154842 910246 085592 505576 > 16⁸⁴ [i]