MinT - Best Known (31−20, 31, s)-Nets in Base 49

Best Known (31−20, 31, s)-Nets in Base 49

(31−20, 31, 101)-Net over F₄₉ — Constructive and digital

Digital (11, 31, 101)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 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, 21, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(31−20, 31, 114)-Net over F₄₉ — Digital

Digital (11, 31, 114)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49³¹, 114, F₄₉, 3, 20) (dual of [(114, 3), 311, 21]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹⁰, 50, F₄₉, 3, 10) (dual of [(50, 3), 140, 11]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(3;140,49) [i]
  2. linear OOA(49²¹, 64, F₄₉, 3, 20) (dual of [(64, 3), 171, 21]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,171P) [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]

(31−20, 31, 16376)-Net in Base 49 — Upper bound on s

There is no (11, 31, 16377)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 24902 396957 000501 122982 039322 656226 246947 990568 676321 > 49³¹ [i]