MinT - Best Known (98−59, 98, s)-Nets in Base 16

Best Known (98−59, 98, s)-Nets in Base 16

(98−59, 98, 110)-Net over F₁₆ — Constructive and digital

Digital (39, 98, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 33, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 65, 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]

(98−59, 98, 128)-Net in Base 16 — Constructive

(39, 98, 128)-net in base 16, using

4 times m-reduction [i] based on (39, 102, 128)-net in base 16, using
- base change [i] based on digital (5, 68, 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]

(98−59, 98, 208)-Net over F₁₆ — Digital

Digital (39, 98, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 98, 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]

(98−59, 98, 8275)-Net in Base 16 — Upper bound on s

There is no (39, 98, 8276)-net in base 16, because

1 times m-reduction [i] would yield (39, 97, 8276)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 632 578241 248559 659577 424648 316487 855713 375318 784925 394931 513382 399960 170371 460349 188226 288782 881925 040212 973421 715536 > 16⁹⁷ [i]