MinT - Best Known (18, 18+28, s)-Nets in Base 128

Best Known (18, 18+28, s)-Nets in Base 128

(18, 18+28, 342)-Net over F₁₂₈ — Constructive and digital

Digital (18, 46, 342)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 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. digital (3, 31, 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]

(18, 18+28, 386)-Net in Base 128 — Constructive

(18, 46, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 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. (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]

(18, 18+28, 388)-Net over F₁₂₈ — Digital

Digital (18, 46, 388)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128⁴⁶, 388, F₁₂₈, 2, 28) (dual of [(388, 2), 730, 29]-NRT-code), using
- constructionÂ X applied to AG(2;F,741P) âŠ‚ AG(2;F,745P) [i] based on
  1. linear OOA(128⁴³, 385, F₁₂₈, 2, 28) (dual of [(385, 2), 727, 29]-NRT-code), using algebraic-geometric NRT-code AG(2;F,741P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]
  2. linear OOA(128³⁹, 385, F₁₂₈, 2, 24) (dual of [(385, 2), 731, 25]-NRT-code), using algebraic-geometric NRT-code AG(2;F,745P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386 (see above)
  3. linear OOA(128³, 3, F₁₂₈, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(128³, 128, F₁₂₈, 2, 3) (dual of [(128, 2), 253, 4]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;253,128) [i]

(18, 18+28, 513)-Net in Base 128

(18, 46, 513)-net in base 128, using

t-expansion [i] based on (17, 46, 513)-net in base 128, using
- 26 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
  - base change [i] based on digital (8, 63, 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]

(18, 18+28, 399334)-Net in Base 128 — Upper bound on s

There is no (18, 46, 399335)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 8 544126 070468 464855 292922 062528 588432 400577 107428 524603 350175 529422 988039 835137 872893 980284 600396 > 128⁴⁶ [i]