MinT - Best Known (39, 39+81, s)-Nets in Base 16

Best Known (39, 39+81, s)-Nets in Base 16

(39, 39+81, 65)-Net over F₁₆ — Constructive and digital

Digital (39, 120, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 120, 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+81, 120)-Net in Base 16 — Constructive

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

t-expansion [i] based on (37, 120, 120)-net in base 16, using
- 10 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+81, 208)-Net over F₁₆ — Digital

Digital (39, 120, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 120, 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+81, 3995)-Net in Base 16 — Upper bound on s

There is no (39, 120, 3996)-net in base 16, because

1 times m-reduction [i] would yield (39, 119, 3996)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 196332 413070 259410 214794 272250 939586 940711 223168 988513 218137 721006 413380 792571 353347 495808 389230 897458 958150 144445 972202 798897 521754 001522 267476 > 16¹¹⁹ [i]