MinT - Best Known (20, 45, s)-Nets in Base 128

Best Known (20, 45, s)-Nets in Base 128

(20, 45, 408)-Net over F₁₂₈ — Constructive and digital

Digital (20, 45, 408)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 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 (5, 30, 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]

(20, 45, 516)-Net in Base 128 — Constructive

(20, 45, 516)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 15, 258)-net in base 128, using
  - 1 times m-reduction [i] based on (3, 16, 258)-net in base 128, using
    - base change [i] based on digital (1, 14, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. (5, 30, 258)-net in base 128, using
  - 2 times m-reduction [i] based on (5, 32, 258)-net in base 128, using
    - base change [i] based on digital (1, 28, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(20, 45, 702)-Net over F₁₂₈ — Digital

Digital (20, 45, 702)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(20, 45, 2218161)-Net in Base 128 — Upper bound on s

There is no (20, 45, 2218162)-net in base 128, because

1 times m-reduction [i] would yield (20, 44, 2218162)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 521 482769 613676 516197 091107 741622 624102 787652 074493 849009 840524 780663 261085 968524 424659 809492 > 128⁴⁴ [i]