MinT - Best Known (75−27, 75, s)-Nets in Base 16

Best Known (75−27, 75, s)-Nets in Base 16

(75−27, 75, 581)-Net over F₁₆ — Constructive and digital

Digital (48, 75, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 19, 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]
2. digital (29, 56, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 28, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(75−27, 75, 594)-Net in Base 16 — Constructive

(48, 75, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (8, 21, 80)-net in base 16, using
  - base change [i] based on digital (1, 14, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (27, 54, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 27, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(75−27, 75, 2105)-Net over F₁₆ — Digital

Digital (48, 75, 2105)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−27, 75, 2701066)-Net in Base 16 — Upper bound on s

There is no (48, 75, 2701067)-net in base 16, because

1 times m-reduction [i] would yield (48, 74, 2701067)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 127315 138014 944562 031403 226473 419650 473617 165467 846500 322989 788586 010536 749072 753419 289216 > 16⁷⁴ [i]