MinT - Best Known (101−57, 101, s)-Nets in Base 25

Best Known (101−57, 101, s)-Nets in Base 25

(101−57, 101, 288)-Net over F₂₅ — Constructive and digital

Digital (44, 101, 288)-net over F₂₅, using

t-expansion [i] based on digital (41, 101, 288)-net over F₂₅, using
- net from sequence [i] based on digital (41, 287)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 41 and N(F) ≥ 288, using
    - a function field by GarcÃa and Quoos [i]

(101−57, 101, 290)-Net over F₂₅ — Digital

Digital (44, 101, 290)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25¹⁰¹, 290, F₂₅, 2, 57) (dual of [(290, 2), 479, 58]-NRT-code), using
- constructionÂ X applied to AG(2;F,516P) âŠ‚ AG(2;F,520P) [i] based on
  1. linear OOA(25⁹⁸, 287, F₂₅, 2, 57) (dual of [(287, 2), 476, 58]-NRT-code), using algebraic-geometric NRT-code AG(2;F,516P) [i] based on function field F/F₂₅ with g(F) = 41 and N(F) ≥ 288, using
    - a function field by GarcÃa and Quoos [i]
  2. linear OOA(25⁹⁴, 287, F₂₅, 2, 53) (dual of [(287, 2), 480, 54]-NRT-code), using algebraic-geometric NRT-code AG(2;F,520P) [i] based on function field F/F₂₅ with g(F) = 41 and N(F) ≥ 288 (see above)
  3. linear OOA(25³, 3, F₂₅, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(25³, 25, F₂₅, 2, 3) (dual of [(25, 2), 47, 4]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;47,25) [i]

(101−57, 101, 46269)-Net in Base 25 — Upper bound on s

There is no (44, 101, 46270)-net in base 25, because

1 times m-reduction [i] would yield (44, 100, 46270)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 62 231976 960432 675227 074639 742894 920314 777441 627892 412377 369204 967629 319822 393184 820332 871242 911129 252121 960628 932757 385015 276903 008868 169025 > 25¹⁰⁰ [i]