MinT - Best Known (101−63, 101, s)-Nets in Base 16

Best Known (101−63, 101, s)-Nets in Base 16

(101−63, 101, 89)-Net over F₁₆ — Constructive and digital

Digital (38, 101, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 32, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 69, 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]

(101−63, 101, 120)-Net in Base 16 — Constructive

(38, 101, 120)-net in base 16, using

t-expansion [i] based on (37, 101, 120)-net in base 16, using
- 29 times m-reduction [i] based on (37, 130, 120)-net in base 16, using
  - base change [i] based on digital (11, 104, 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]

(101−63, 101, 208)-Net over F₁₆ — Digital

Digital (38, 101, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 101, 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]

(101−63, 101, 6324)-Net in Base 16 — Upper bound on s

There is no (38, 101, 6325)-net in base 16, because

1 times m-reduction [i] would yield (38, 100, 6325)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 587841 749941 980958 285751 110603 465795 029514 267181 333075 755406 681189 443135 372751 077236 338943 902129 764279 889860 059233 497376 > 16¹⁰⁰ [i]