MinT - Best Known (83−45, 83, s)-Nets in Base 16

Best Known (83−45, 83, s)-Nets in Base 16

(83−45, 83, 130)-Net over F₁₆ — Constructive and digital

Digital (38, 83, 130)-net over F₁₆, using

7 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)

(83−45, 83, 177)-Net in Base 16 — Constructive

(38, 83, 177)-net in base 16, using

10 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]

(83−45, 83, 208)-Net over F₁₆ — Digital

Digital (38, 83, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 83, 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]

(83−45, 83, 225)-Net in Base 16

(38, 83, 225)-net in base 16, using

1 times m-reduction [i] based on (38, 84, 225)-net in base 16, using
- base change [i] based on digital (10, 56, 225)-net over F₆₄, using
  - net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
      - a curve from the manYPoints database [i]

(83−45, 83, 18559)-Net in Base 16 — Upper bound on s

There is no (38, 83, 18560)-net in base 16, because

1 times m-reduction [i] would yield (38, 82, 18560)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 547 159313 248918 643868 672986 804530 440367 087305 902418 515288 932804 001545 410968 696361 869986 144567 657801 > 16⁸² [i]