MinT - Best Known (16, 48, s)-Nets in Base 49

Best Known (16, 48, s)-Nets in Base 49

(16, 48, 100)-Net over F₄₉ — Constructive and digital

Digital (16, 48, 100)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 16, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
      - the rational function field F₄₉(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 32, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(16, 48, 104)-Net over F₄₉ — Digital

Digital (16, 48, 104)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49⁴⁸, 104, F₄₉, 2, 32) (dual of [(104, 2), 160, 33]-NRT-code), using
- constructionÂ X applied to AG(2;F,149P) âŠ‚ AG(2;F,163P) [i] based on
  1. linear OOA(49³⁵, 91, F₄₉, 2, 32) (dual of [(91, 2), 147, 33]-NRT-code), using algebraic-geometric NRT-code AG(2;F,149P) [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(49²¹, 91, F₄₉, 2, 18) (dual of [(91, 2), 161, 19]-NRT-code), using algebraic-geometric NRT-code AG(2;F,163P) [i] based on function field F/F₄₉ with g(F) = 3 and N(F) ≥ 92 (see above)
  3. linear OOA(49¹³, 13, F₄₉, 2, 13) (dual of [(13, 2), 13, 14]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(49¹³, 49, F₄₉, 2, 13) (dual of [(49, 2), 85, 14]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;85,49) [i]

(16, 48, 16660)-Net in Base 49 — Upper bound on s

There is no (16, 48, 16661)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 1348 103208 669980 632710 629978 666595 411915 740388 478029 014295 170723 092500 995770 992385 > 49⁴⁸ [i]