MinT - Best Known (89−52, 89, s)-Nets in Base 16

Best Known (89−52, 89, s)-Nets in Base 16

(89−52, 89, 114)-Net over F₁₆ — Constructive and digital

Digital (37, 89, 114)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 31, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (6, 58, 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]

(89−52, 89, 177)-Net in Base 16 — Constructive

(37, 89, 177)-net in base 16, using

1 times m-reduction [i] based on (37, 90, 177)-net in base 16, using
- base change [i] based on digital (7, 60, 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]

(89−52, 89, 208)-Net over F₁₆ — Digital

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]

(89−52, 89, 9296)-Net in Base 16 — Upper bound on s

There is no (37, 89, 9297)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146870 473976 322633 358564 643210 989522 977398 118052 959223 221951 899517 431830 883060 322543 327178 201333 754833 351456 > 16⁸⁹ [i]