MinT - Best Known (32, 32+85, s)-Nets in Base 16

Best Known (32, 32+85, s)-Nets in Base 16

(32, 32+85, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 117, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 117, 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]

(32, 32+85, 98)-Net in Base 16 — Constructive

(32, 117, 98)-net in base 16, using

8 times m-reduction [i] based on (32, 125, 98)-net in base 16, using
- base change [i] based on digital (7, 100, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(32, 32+85, 168)-Net over F₁₆ — Digital

Digital (32, 117, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 117, 168)-net over F₁₆, using
- net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
    - a curve from the manYPoints database [i]

(32, 32+85, 2306)-Net in Base 16 — Upper bound on s

There is no (32, 117, 2307)-net in base 16, because

1 times m-reduction [i] would yield (32, 116, 2307)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 47 752997 809273 935233 709717 313889 042854 378310 259976 592754 760971 201390 246503 817266 700546 062042 874031 185625 955924 917133 906875 747453 857095 098336 > 16¹¹⁶ [i]