MinT - Best Known (53−31, 53, s)-Nets in Base 128

Best Known (53−31, 53, s)-Nets in Base 128

(53−31, 53, 384)-Net over F₁₂₈ — Constructive and digital

Digital (22, 53, 384)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (22, 54, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 19, 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 (3, 35, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(53−31, 53, 408)-Net in Base 128 — Constructive

(22, 53, 408)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. (6, 37, 258)-net in base 128, using
  - 3 times m-reduction [i] based on (6, 40, 258)-net in base 128, using
    - base change [i] based on digital (1, 35, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(53−31, 53, 516)-Net over F₁₂₈ — Digital

Digital (22, 53, 516)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(53−31, 53, 1020832)-Net in Base 128 — Upper bound on s

There is no (22, 53, 1020833)-net in base 128, because

1 times m-reduction [i] would yield (22, 52, 1020833)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 37 577003 016962 169769 437346 110968 259798 695378 631268 138046 248647 668598 753563 413147 164608 879955 757681 801399 308576 > 128⁵² [i]