MinT - Best Known (39, 118, s)-Nets in Base 16

Best Known (39, 118, s)-Nets in Base 16

(39, 118, 65)-Net over F₁₆ — Constructive and digital

Digital (39, 118, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 118, 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, 118, 120)-Net in Base 16 — Constructive

(39, 118, 120)-net in base 16, using

t-expansion [i] based on (37, 118, 120)-net in base 16, using
- 12 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, 118, 208)-Net over F₁₆ — Digital

Digital (39, 118, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 118, 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, 118, 4182)-Net in Base 16 — Upper bound on s

There is no (39, 118, 4183)-net in base 16, because

1 times m-reduction [i] would yield (39, 117, 4183)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 764 057042 246063 396622 939473 937053 314906 821393 046509 723214 093041 777137 716292 176381 068790 403917 831202 587392 871137 607319 084735 796535 599300 686656 > 16¹¹⁷ [i]