MinT - Best Known (17, 17+33, s)-Nets in Base 49

Best Known (17, 17+33, s)-Nets in Base 49

(17, 17+33, 101)-Net over F₄₉ — Constructive and digital

Digital (17, 50, 101)-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 (1, 34, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(17, 17+33, 114)-Net over F₄₉ — Digital

Digital (17, 50, 114)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49⁵⁰, 114, F₄₉, 2, 33) (dual of [(114, 2), 178, 34]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹⁶, 50, F₄₉, 2, 16) (dual of [(50, 2), 84, 17]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(2;84,49) [i]
  2. linear OOA(49³⁴, 64, F₄₉, 2, 33) (dual of [(64, 2), 94, 34]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,94P) [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]

(17, 17+33, 21250)-Net in Base 49 — Upper bound on s

There is no (17, 50, 21251)-net in base 49, because

1 times m-reduction [i] would yield (17, 49, 21251)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 66049 697922 749864 451210 947464 560269 229983 593457 600112 545911 542377 860655 547872 467201 > 49⁴⁹ [i]