MinT - Best Known (27, 68, s)-Nets in Base 128

Best Known (27, 68, s)-Nets in Base 128

(27, 68, 384)-Net over F₁₂₈ — Constructive and digital

Digital (27, 68, 384)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (27, 69, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 24, 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 (3, 45, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(27, 68, 407)-Net in Base 128 — Constructive

(27, 68, 407)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. 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]
2. (6, 47, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (6, 48, 257)-net in base 128, using
    - base change [i] based on digital (0, 42, 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]

(27, 68, 515)-Net over F₁₂₈ — Digital

Digital (27, 68, 515)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128⁶⁸, 515, F₁₂₈, 2, 41) (dual of [(515, 2), 962, 42]-NRT-code), using
- constructionÂ X applied to AG(2;F,982P) âŠ‚ AG(2;F,986P) [i] based on
  1. linear OOA(128⁶⁵, 512, F₁₂₈, 2, 41) (dual of [(512, 2), 959, 42]-NRT-code), using algebraic-geometric NRT-code AG(2;F,982P) [i] based on function field F/F₁₂₈ with g(F) = 24 and N(F) ≥ 513, using
    - a curve from the manYPoints database [i]
  2. linear OOA(128⁶¹, 512, F₁₂₈, 2, 37) (dual of [(512, 2), 963, 38]-NRT-code), using algebraic-geometric NRT-code AG(2;F,986P) [i] based on function field F/F₁₂₈ with g(F) = 24 and N(F) ≥ 513 (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]

(27, 68, 749290)-Net in Base 128 — Upper bound on s

There is no (27, 68, 749291)-net in base 128, because

1 times m-reduction [i] would yield (27, 67, 749291)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1524 305706 310495 036817 176262 872118 918379 536328 714746 684838 925020 101345 757877 258703 333774 922238 624390 459500 779350 091852 883944 481134 291549 078308 > 128⁶⁷ [i]