MinT - Best Known (104−65, 104, s)-Nets in Base 16

Best Known (104−65, 104, s)-Nets in Base 16

(104−65, 104, 89)-Net over F₁₆ — Constructive and digital

Digital (39, 104, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 33, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 71, 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]

(104−65, 104, 120)-Net in Base 16 — Constructive

(39, 104, 120)-net in base 16, using

t-expansion [i] based on (37, 104, 120)-net in base 16, using
- 26 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]

(104−65, 104, 208)-Net over F₁₆ — Digital

Digital (39, 104, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 104, 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]

(104−65, 104, 6387)-Net in Base 16 — Upper bound on s

There is no (39, 104, 6388)-net in base 16, because

1 times m-reduction [i] would yield (39, 103, 6388)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 10585 432216 927351 131663 513001 083338 118974 662337 575838 476176 910673 875396 138023 638339 984683 103065 422243 795687 638578 280831 009341 > 16¹⁰³ [i]