MinT - Best Known (70−37, 70, s)-Nets in Base 128

Best Known (70−37, 70, s)-Nets in Base 128

(70−37, 70, 504)-Net over F₁₂₈ — Constructive and digital

Digital (33, 70, 504)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (33, 71, 504)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 24, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]
  2. digital (9, 47, 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]

(70−37, 70, 548)-Net in Base 128 — Constructive

(33, 70, 548)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (6, 24, 260)-net in base 128, using
  - base change [i] based on digital (3, 21, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
2. digital (9, 46, 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]

(70−37, 70, 1425)-Net over F₁₂₈ — Digital

Digital (33, 70, 1425)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(70−37, 70, 7111546)-Net in Base 128 — Upper bound on s

There is no (33, 70, 7111547)-net in base 128, because

1 times m-reduction [i] would yield (33, 69, 7111547)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 24 974050 150997 498041 630158 672673 929750 631497 580707 420997 686777 827422 067546 126946 397677 363169 971406 984623 460798 446061 937303 892002 681007 681115 983881 > 128⁶⁹ [i]