MinT - Best Known (37−20, 37, s)-Nets in Base 128

Best Known (37−20, 37, s)-Nets in Base 128

(37−20, 37, 408)-Net over F₁₂₈ — Constructive and digital

Digital (17, 37, 408)-net over F₁₂₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 6, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 10, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
3. digital (1, 21, 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]

(37−20, 37, 516)-Net in Base 128 — Constructive

(17, 37, 516)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 13, 258)-net in base 128, using
  - 3 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. (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 (see above)

(37−20, 37, 802)-Net over F₁₂₈ — Digital

Digital (17, 37, 802)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(37−20, 37, 2232793)-Net in Base 128 — Upper bound on s

There is no (17, 37, 2232794)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 926338 356430 142272 681960 168953 695657 215400 011471 550056 986488 598640 302812 128632 > 128³⁷ [i]