MinT - Best Known (14−9, 14, s)-Nets in Base 49

Best Known (14−9, 14, s)-Nets in Base 49

(14−9, 14, 101)-Net over F₄₉ — Constructive and digital

Digital (5, 14, 101)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 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 (1, 10, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(14−9, 14, 114)-Net over F₄₉ — Digital

Digital (5, 14, 114)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49¹⁴, 114, F₄₉, 2, 9) (dual of [(114, 2), 214, 10]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49⁴, 50, F₄₉, 2, 4) (dual of [(50, 2), 96, 5]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(2;96,49) [i]
  2. linear OOA(49¹⁰, 64, F₄₉, 2, 9) (dual of [(64, 2), 118, 10]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,118P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(14−9, 14, 14351)-Net in Base 49 — Upper bound on s

There is no (5, 14, 14352)-net in base 49, because

1 times m-reduction [i] would yield (5, 13, 14352)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 9388 474729 647035 563009 > 49¹³ [i]