MinT - Best Known (120−43, 120, s)-Nets in Base 16

Best Known (120−43, 120, s)-Nets in Base 16

(120−43, 120, 585)-Net over F₁₆ — Constructive and digital

Digital (77, 120, 585)-net over F₁₆, using

16¹ times duplication [i] based on digital (76, 119, 585)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 27, 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 (49, 92, 520)-net over F₁₆, using
    - trace code for nets [i] based on digital (3, 46, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(120−43, 120, 594)-Net in Base 16 — Constructive

(77, 120, 594)-net in base 16, using

16¹ times duplication [i] based on (76, 119, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (12, 33, 80)-net in base 16, using
    - base change [i] based on digital (1, 22, 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 (43, 86, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 43, 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]

(120−43, 120, 3055)-Net over F₁₆ — Digital

Digital (77, 120, 3055)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(120−43, 120, 3852534)-Net in Base 16 — Upper bound on s

There is no (77, 120, 3852535)-net in base 16, because

1 times m-reduction [i] would yield (77, 119, 3852535)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 195109 298626 196282 490932 470545 651600 923008 177534 779177 275546 186877 412378 950354 579063 700879 419211 575199 435133 393557 963297 975900 185500 978904 978526 > 16¹¹⁹ [i]