MinT - Best Known (39, 39+87, s)-Nets in Base 16

Best Known (39, 39+87, s)-Nets in Base 16

(39, 39+87, 65)-Net over F₁₆ — Constructive and digital

Digital (39, 126, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 126, 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]

(39, 39+87, 120)-Net in Base 16 — Constructive

(39, 126, 120)-net in base 16, using

t-expansion [i] based on (37, 126, 120)-net in base 16, using
- 4 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]

(39, 39+87, 208)-Net over F₁₆ — Digital

Digital (39, 126, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 126, 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]

(39, 39+87, 3538)-Net in Base 16 — Upper bound on s

There is no (39, 126, 3539)-net in base 16, because

1 times m-reduction [i] would yield (39, 125, 3539)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 295730 114166 195630 769614 600566 161574 413706 082740 501350 174905 239178 707566 627320 128416 950056 869215 344279 591700 490979 375184 152494 470148 241673 898169 786656 > 16¹²⁵ [i]