MinT - Best Known (42−23, 42, s)-Nets in Base 128

Best Known (42−23, 42, s)-Nets in Base 128

(42−23, 42, 408)-Net over F₁₂₈ — Constructive and digital

Digital (19, 42, 408)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 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, 28, 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]

(42−23, 42, 516)-Net in Base 128 — Constructive

(19, 42, 516)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 14, 258)-net in base 128, using
  - 2 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, 28, 258)-net in base 128, using
  - 4 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)

(42−23, 42, 762)-Net over F₁₂₈ — Digital

Digital (19, 42, 762)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−23, 42, 2762840)-Net in Base 128 — Upper bound on s

There is no (19, 42, 2762841)-net in base 128, because

1 times m-reduction [i] would yield (19, 41, 2762841)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 248 661656 368891 246399 973930 770181 891108 077758 732812 767882 513913 287141 308039 207517 051056 > 128⁴¹ [i]