MinT - Best Known (31, 68, s)-Nets in Base 128

Best Known (31, 68, s)-Nets in Base 128

(31, 68, 480)-Net over F₁₂₈ — Constructive and digital

Digital (31, 68, 480)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (31, 69, 480)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 22, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, 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]

(31, 68, 546)-Net in Base 128 — Constructive

(31, 68, 546)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 22, 258)-net in base 128, using
  - 2 times m-reduction [i] based on (4, 24, 258)-net in base 128, using
    - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-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]

(31, 68, 1093)-Net over F₁₂₈ — Digital

Digital (31, 68, 1093)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 68, 4147908)-Net in Base 128 — Upper bound on s

There is no (31, 68, 4147909)-net in base 128, because

1 times m-reduction [i] would yield (31, 67, 4147909)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1524 292112 319207 539447 938699 580781 878528 400154 843252 738756 264184 439633 265123 294609 161292 665080 419895 764097 013033 372523 515403 066176 184702 479160 > 128⁶⁷ [i]