MinT - Best Known (90−52, 90, s)-Nets in Base 16

Best Known (90−52, 90, s)-Nets in Base 16

(90−52, 90, 130)-Net over F₁₆ — Constructive and digital

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)

(90−52, 90, 177)-Net in Base 16 — Constructive

(38, 90, 177)-net in base 16, using

3 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]

(90−52, 90, 208)-Net over F₁₆ — Digital

Digital (38, 90, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 90, 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]

(90−52, 90, 10344)-Net in Base 16 — Upper bound on s

There is no (38, 90, 10345)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 351074 102702 579089 419755 625004 458143 387962 302099 662165 884644 291608 208769 334589 350352 611973 300460 388482 500176 > 16⁹⁰ [i]