MinT - Best Known (102−63, 102, s)-Nets in Base 16

Best Known (102−63, 102, s)-Nets in Base 16

(102−63, 102, 98)-Net over F₁₆ — Constructive and digital

Digital (39, 102, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 33, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (6, 69, 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]

(102−63, 102, 128)-Net in Base 16 — Constructive

(39, 102, 128)-net in base 16, using

base change [i] based on digital (5, 68, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(102−63, 102, 208)-Net over F₁₆ — Digital

Digital (39, 102, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 102, 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]

(102−63, 102, 6917)-Net in Base 16 — Upper bound on s

There is no (39, 102, 6918)-net in base 16, because

1 times m-reduction [i] would yield (39, 101, 6918)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 41 338686 415230 538055 629210 103034 585393 791300 659397 006555 323044 791210 138453 066161 525417 680099 464749 781900 980336 279991 226496 > 16¹⁰¹ [i]