MinT - Best Known (8, 22, s)-Nets in Base 49

Best Known (8, 22, s)-Nets in Base 49

(8, 22, 101)-Net over F₄₉ — Constructive and digital

Digital (8, 22, 101)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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, 15, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(8, 22, 114)-Net over F₄₉ — Digital

Digital (8, 22, 114)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49²², 114, F₄₉, 2, 14) (dual of [(114, 2), 206, 15]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49⁷, 50, F₄₉, 2, 7) (dual of [(50, 2), 93, 8]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(2;93,49) [i]
  2. linear OOA(49¹⁵, 64, F₄₉, 2, 14) (dual of [(64, 2), 113, 15]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,113P) [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]

(8, 22, 14442)-Net in Base 49 — Upper bound on s

There is no (8, 22, 14443)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 15 294034 047606 312968 678439 954273 384049 > 49²² [i]