MinT - Best Known (130−91, 130, s)-Nets in Base 16

Best Known (130−91, 130, s)-Nets in Base 16

(130−91, 130, 65)-Net over F₁₆ — Constructive and digital

Digital (39, 130, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 130, 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]

(130−91, 130, 120)-Net in Base 16 — Constructive

(39, 130, 120)-net in base 16, using

t-expansion [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]

(130−91, 130, 208)-Net over F₁₆ — Digital

Digital (39, 130, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 130, 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]

(130−91, 130, 3300)-Net in Base 16 — Upper bound on s

There is no (39, 130, 3301)-net in base 16, because

1 times m-reduction [i] would yield (39, 129, 3301)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 215041 882421 698696 361739 390973 776763 257412 178784 778978 365982 404865 834423 039427 916884 516099 760736 893023 269420 116441 697681 633644 805765 853643 450746 392102 442176 > 16¹²⁹ [i]