MinT - Best Known (53−36, 53, s)-Nets in Base 49

Best Known (53−36, 53, s)-Nets in Base 49

(53−36, 53, 96)-Net over F₄₉ — Constructive and digital

Digital (17, 53, 96)-net over F₄₉, using

net from sequence [i] based on digital (17, 95)-sequence over F₄₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) ≤ 17 and N(F) ≥ 96, using
  - F₄ from the tower of function fields by GarcÃa, Stichtenoth, and RÃ¼ck over F₄₉ [i]

(53−36, 53, 100)-Net over F₄₉ — Digital

Digital (17, 53, 100)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49⁵³, 100, F₄₉, 3, 36) (dual of [(100, 3), 247, 37]-NRT-code), using
- strength reduction [i] based on linear OOA(49⁵³, 100, F₄₉, 3, 37) (dual of [(100, 3), 247, 38]-NRT-code), using
  - constructionÂ X applied to AG(3;F,235P) âŠ‚ AG(3;F,249P) [i] based on
    1. linear OOA(49⁴⁰, 91, F₄₉, 3, 37) (dual of [(91, 3), 233, 38]-NRT-code), using algebraic-geometric NRT-code AG(3;F,235P) [i] based on function field F/F₄₉ with g(F) = 3 and N(F) ≥ 92, using
      - a curve from the manYPoints database [i]
    2. linear OOA(49²⁶, 91, F₄₉, 3, 23) (dual of [(91, 3), 247, 24]-NRT-code), using algebraic-geometric NRT-code AG(3;F,249P) [i] based on function field F/F₄₉ with g(F) = 3 and N(F) ≥ 92 (see above)
    3. linear OOA(49¹³, 9, F₄₉, 3, 13) (dual of [(9, 3), 14, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(49¹³, 49, F₄₉, 3, 13) (dual of [(49, 3), 134, 14]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;134,49) [i]

(53−36, 53, 14904)-Net in Base 49 — Upper bound on s

There is no (17, 53, 14905)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 380740 996876 796503 209211 455236 393645 797793 724738 202885 284562 370547 370866 195354 960233 747809 > 49⁵³ [i]