MinT - Best Known (72−39, 72, s)-Nets in Base 128

Best Known (72−39, 72, s)-Nets in Base 128

(72−39, 72, 504)-Net over F₁₂₈ — Constructive and digital

Digital (33, 72, 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, 48, 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]

(72−39, 72, 547)-Net in Base 128 — Constructive

(33, 72, 547)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (5, 24, 259)-net in base 128, using
  - base change [i] based on digital (2, 21, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
2. digital (9, 48, 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]

(72−39, 72, 1182)-Net over F₁₂₈ — Digital

Digital (33, 72, 1182)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−39, 72, 4674356)-Net in Base 128 — Upper bound on s

There is no (33, 72, 4674357)-net in base 128, because

1 times m-reduction [i] would yield (33, 71, 4674357)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 409173 891242 901828 018777 315869 613582 963456 634170 982641 114032 693072 832287 365190 589141 683230 724592 585183 428713 771040 752886 193238 673856 467485 442203 227892 > 128⁷¹ [i]