MinT - Best Known (8, 8+15, s)-Nets in Base 49

Best Known (8, 8+15, s)-Nets in Base 49

(8, 8+15, 101)-Net over F₄₉ — Constructive and digital

Digital (8, 23, 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, 16, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(8, 8+15, 114)-Net over F₄₉ — Digital

Digital (8, 23, 114)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49²³, 114, F₄₉, 3, 15) (dual of [(114, 3), 319, 16]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49⁷, 50, F₄₉, 3, 7) (dual of [(50, 3), 143, 8]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(3;143,49) [i]
  2. linear OOA(49¹⁶, 64, F₄₉, 3, 15) (dual of [(64, 3), 176, 16]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,176P) [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, 8+15, 14442)-Net in Base 49 — Upper bound on s

There is no (8, 23, 14443)-net in base 49, because

1 times m-reduction [i] would yield (8, 22, 14443)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 15 294034 047606 312968 678439 954273 384049 > 49²² [i]