MinT - Best Known (23, 23+39, s)-Nets in Base 128

Best Known (23, 23+39, s)-Nets in Base 128

(23, 23+39, 342)-Net over F₁₂₈ — Constructive and digital

Digital (23, 62, 342)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 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, 42, 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]

(23, 23+39, 389)-Net over F₁₂₈ — Digital

Digital (23, 62, 389)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128⁶², 389, F₁₂₈, 4, 39) (dual of [(389, 4), 1494, 40]-NRT-code), using
- constructionÂ X applied to AG(4;F,1500P) âŠ‚ AG(4;F,1509P) [i] based on
  1. linear OOA(128⁵⁴, 385, F₁₂₈, 4, 39) (dual of [(385, 4), 1486, 40]-NRT-code), using algebraic-geometric NRT-code AG(4;F,1500P) [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(128⁴⁵, 385, F₁₂₈, 4, 30) (dual of [(385, 4), 1495, 31]-NRT-code), using algebraic-geometric NRT-code AG(4;F,1509P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386 (see above)
  3. linear OOA(128⁸, 4, F₁₂₈, 4, 8) (dual of [(4, 4), 8, 9]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(128⁸, 128, F₁₂₈, 4, 8) (dual of [(128, 4), 504, 9]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(4;504,128) [i]

(23, 23+39, 513)-Net in Base 128

(23, 62, 513)-net in base 128, using

t-expansion [i] based on (17, 62, 513)-net in base 128, using
- 10 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]

(23, 23+39, 363625)-Net in Base 128 — Upper bound on s

There is no (23, 62, 363626)-net in base 128, because

1 times m-reduction [i] would yield (23, 61, 363626)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 346 599031 214666 682262 809277 476799 237746 204380 959167 095237 164067 784230 894309 324207 424943 846014 578217 045095 063582 545667 057468 400792 > 128⁶¹ [i]