MinT - Best Known (123−44, 123, s)-Nets in Base 16

Best Known (123−44, 123, s)-Nets in Base 16

(123−44, 123, 585)-Net over F₁₆ — Constructive and digital

Digital (79, 123, 585)-net over F₁₆, using

1 times m-reduction [i] based on digital (79, 124, 585)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 28, 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 (51, 96, 520)-net over F₁₆, using
    - trace code for nets [i] based on digital (3, 48, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(123−44, 123, 594)-Net in Base 16 — Constructive

(79, 123, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (13, 35, 80)-net in base 16, using
  - 1 times m-reduction [i] based on (13, 36, 80)-net in base 16, using
    - base change [i] based on digital (1, 24, 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 (44, 88, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 44, 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]

(123−44, 123, 3153)-Net over F₁₆ — Digital

Digital (79, 123, 3153)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−44, 123, 3257508)-Net in Base 16 — Upper bound on s

There is no (79, 123, 3257509)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 12786 743098 270405 162229 026755 500064 191270 877007 089902 626801 010344 923360 712510 064091 330517 887135 053415 131443 713897 022785 896146 110610 078479 888173 698096 > 16¹²³ [i]