MinT - Best Known (77−41, 77, s)-Nets in Base 128

Best Known (77−41, 77, s)-Nets in Base 128

(77−41, 77, 504)-Net over F₁₂₈ — Constructive and digital

Digital (36, 77, 504)-net over F₁₂₈, using

t-expansion [i] based on digital (35, 77, 504)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 26, 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, 51, 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]

(77−41, 77, 548)-Net in Base 128 — Constructive

(36, 77, 548)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (7, 27, 260)-net in base 128, using
  - 5 times m-reduction [i] based on (7, 32, 260)-net in base 128, using
    - base change [i] based on digital (3, 28, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (9, 50, 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]

(77−41, 77, 1434)-Net over F₁₂₈ — Digital

Digital (36, 77, 1434)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−41, 77, 6651196)-Net in Base 128 — Upper bound on s

There is no (36, 77, 6651197)-net in base 128, because

1 times m-reduction [i] would yield (36, 76, 6651197)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 14059 136242 830719 063699 065251 736075 920285 062132 811411 146632 353304 808835 570347 864557 255701 978197 274557 638918 474512 777971 165840 301941 929176 121108 306834 882145 587111 > 128⁷⁶ [i]