MinT - Best Known (51−31, 51, s)-Nets in Base 128

Best Known (51−31, 51, s)-Nets in Base 128

(51−31, 51, 345)-Net over F₁₂₈ — Constructive and digital

Digital (20, 51, 345)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 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, 36, 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]

(51−31, 51, 386)-Net in Base 128 — Constructive

(20, 51, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 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. (5, 36, 257)-net in base 128, using
  - 4 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
    - base change [i] based on digital (0, 35, 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]

(51−31, 51, 390)-Net over F₁₂₈ — Digital

Digital (20, 51, 390)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128⁵¹, 390, F₁₂₈, 2, 31) (dual of [(390, 2), 729, 32]-NRT-code), using
- constructionÂ X applied to AG(2;F,738P) âŠ‚ AG(2;F,744P) [i] based on
  1. linear OOA(128⁴⁶, 385, F₁₂₈, 2, 31) (dual of [(385, 2), 724, 32]-NRT-code), using algebraic-geometric NRT-code AG(2;F,738P) [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(128⁴⁰, 385, F₁₂₈, 2, 25) (dual of [(385, 2), 730, 26]-NRT-code), using algebraic-geometric NRT-code AG(2;F,744P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386 (see above)
  3. linear OOA(128⁵, 5, F₁₂₈, 2, 5) (dual of [(5, 2), 5, 6]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(128⁵, 128, F₁₂₈, 2, 5) (dual of [(128, 2), 251, 6]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;251,128) [i]

(51−31, 51, 513)-Net in Base 128

(20, 51, 513)-net in base 128, using

t-expansion [i] based on (17, 51, 513)-net in base 128, using
- 21 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]

(51−31, 51, 534552)-Net in Base 128 — Upper bound on s

There is no (20, 51, 534553)-net in base 128, because

1 times m-reduction [i] would yield (20, 50, 534553)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 2293 530483 479129 519112 122089 547930 691865 344812 857181 227042 071189 637202 456557 004951 237751 374404 083335 518352 > 128⁵⁰ [i]