MinT - Best Known (16, 38, s)-Nets in Base 128

Best Known (16, 38, s)-Nets in Base 128

(16, 38, 345)-Net over F₁₂₈ — Constructive and digital

Digital (16, 38, 345)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 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 (5, 27, 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]

(16, 38, 407)-Net in Base 128 — Constructive

(16, 38, 407)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 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. (4, 26, 257)-net in base 128, using
  - 6 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
    - base change [i] based on digital (0, 28, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(16, 38, 455)-Net over F₁₂₈ — Digital

Digital (16, 38, 455)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(16, 38, 513)-Net in Base 128

(16, 38, 513)-net in base 128, using

26 times m-reduction [i] based on (16, 64, 513)-net in base 128, using
- base change [i] based on digital (8, 56, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(16, 38, 735631)-Net in Base 128 — Upper bound on s

There is no (16, 38, 735632)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 118 572775 513213 262464 636504 859201 819384 948488 114934 962481 566543 261607 605420 110555 > 128³⁸ [i]