MinT - Best Known (56−30, 56, s)-Nets in Base 128

Best Known (56−30, 56, s)-Nets in Base 128

(56−30, 56, 438)-Net over F₁₂₈ — Constructive and digital

Digital (26, 56, 438)-net over F₁₂₈, using

2 times m-reduction [i] based on digital (26, 58, 438)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 17, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (9, 41, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(56−30, 56, 518)-Net in Base 128 — Constructive

(26, 56, 518)-net in base 128, using

base change [i] based on digital (19, 49, 518)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (2, 17, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
  2. digital (2, 32, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(56−30, 56, 1092)-Net over F₁₂₈ — Digital

Digital (26, 56, 1092)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−30, 56, 3722882)-Net in Base 128 — Upper bound on s

There is no (26, 56, 3722883)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 10086 940868 490334 843653 607747 564560 402671 274820 193279 046253 087163 639789 076950 376293 346061 946543 936478 881342 792342 610816 > 128⁵⁶ [i]