MinT - Best Known (89−51, 89, s)-Nets in Base 16

Best Known (89−51, 89, s)-Nets in Base 16

(89−51, 89, 130)-Net over F₁₆ — Constructive and digital

Digital (38, 89, 130)-net over F₁₆, using

1 times m-reduction [i] based on digital (38, 90, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 32, 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, 58, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(89−51, 89, 177)-Net in Base 16 — Constructive

(38, 89, 177)-net in base 16, using

4 times m-reduction [i] based on (38, 93, 177)-net in base 16, using
- base change [i] based on digital (7, 62, 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]

(89−51, 89, 208)-Net over F₁₆ — Digital

Digital (38, 89, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 89, 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]

(89−51, 89, 11736)-Net in Base 16 — Upper bound on s

There is no (38, 89, 11737)-net in base 16, because

1 times m-reduction [i] would yield (38, 88, 11737)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9187 444850 092106 248309 866746 509961 348821 202798 557215 034477 057440 572475 241240 099127 574699 875171 545230 591376 > 16⁸⁸ [i]