MinT - Best Known (128−89, 128, s)-Nets in Base 16

Best Known (128−89, 128, s)-Nets in Base 16

(128−89, 128, 65)-Net over F₁₆ — Constructive and digital

Digital (39, 128, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 128, 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]

(128−89, 128, 120)-Net in Base 16 — Constructive

(39, 128, 120)-net in base 16, using

t-expansion [i] based on (37, 128, 120)-net in base 16, using
- 2 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]

(128−89, 128, 208)-Net over F₁₆ — Digital

Digital (39, 128, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 128, 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]

(128−89, 128, 3414)-Net in Base 16 — Upper bound on s

There is no (39, 128, 3415)-net in base 16, because

1 times m-reduction [i] would yield (39, 127, 3415)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 848 400788 044346 464194 737105 461387 442357 572107 265668 404387 060770 308904 804737 355384 013093 915189 276315 116881 618631 213790 368204 836801 113434 084486 447754 830276 > 16¹²⁷ [i]