MinT - Best Known (84−53, 84, s)-Nets in Base 16

Best Known (84−53, 84, s)-Nets in Base 16

(84−53, 84, 71)-Net over F₁₆ — Constructive and digital

Digital (31, 84, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 28, 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 (3, 56, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(84−53, 84, 120)-Net in Base 16 — Constructive

(31, 84, 120)-net in base 16, using

16 times m-reduction [i] based on (31, 100, 120)-net in base 16, using
- base change [i] based on digital (11, 80, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(84−53, 84, 168)-Net over F₁₆ — Digital

Digital (31, 84, 168)-net over F₁₆, using

net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
  - a curve from the manYPoints database [i]

(84−53, 84, 4896)-Net in Base 16 — Upper bound on s

There is no (31, 84, 4897)-net in base 16, because

1 times m-reduction [i] would yield (31, 83, 4897)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8783 841913 724132 595201 787773 159624 822467 980324 212685 767738 356825 383279 376732 100696 462003 427462 885456 > 16⁸³ [i]