MinT - Best Known (99−61, 99, s)-Nets in Base 16

Best Known (99−61, 99, s)-Nets in Base 16

(99−61, 99, 98)-Net over F₁₆ — Constructive and digital

Digital (38, 99, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 32, 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, 67, 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]

(99−61, 99, 128)-Net in Base 16 — Constructive

(38, 99, 128)-net in base 16, using

base change [i] based on digital (5, 66, 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]

(99−61, 99, 208)-Net over F₁₆ — Digital

Digital (38, 99, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 99, 208)-net over F₁₆, using
- net from sequence [i] based on digital (37, 207)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 37 and N(F) ≥ 208, using
    - Drinfeld modules of rank 1 [i]

(99−61, 99, 6872)-Net in Base 16 — Upper bound on s

There is no (38, 99, 6873)-net in base 16, because

1 times m-reduction [i] would yield (38, 98, 6873)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 10108 578102 055774 336094 440382 780896 247883 633109 136423 240545 423840 293977 829526 682448 525902 901844 680258 452978 027573 346976 > 16⁹⁸ [i]