MinT - Best Known (99−67, 99, s)-Nets in Base 16

Best Known (99−67, 99, s)-Nets in Base 16

(99−67, 99, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 99, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 99, 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]

(99−67, 99, 120)-Net in Base 16 — Constructive

(32, 99, 120)-net in base 16, using

6 times m-reduction [i] based on (32, 105, 120)-net in base 16, using
- base change [i] based on digital (11, 84, 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]

(99−67, 99, 168)-Net over F₁₆ — Digital

Digital (32, 99, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 99, 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]

(99−67, 99, 3286)-Net in Base 16 — Upper bound on s

There is no (32, 99, 3287)-net in base 16, because

1 times m-reduction [i] would yield (32, 98, 3287)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 10133 112360 840170 664233 874454 925761 570955 527094 237559 657497 354105 765831 162837 392568 249115 291257 151171 063962 998927 670716 > 16⁹⁸ [i]