MinT - Best Known (15, 15+28, s)-Nets in Base 49

Best Known (15, 15+28, s)-Nets in Base 49

(15, 15+28, 101)-Net over F₄₉ — Constructive and digital

Digital (15, 43, 101)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 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, 29, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(15, 15+28, 114)-Net over F₄₉ — Digital

Digital (15, 43, 114)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49⁴³, 114, F₄₉, 2, 28) (dual of [(114, 2), 185, 29]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹⁴, 50, F₄₉, 2, 14) (dual of [(50, 2), 86, 15]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(2;86,49) [i]
  2. linear OOA(49²⁹, 64, F₄₉, 2, 28) (dual of [(64, 2), 99, 29]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,99P) [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]

(15, 15+28, 19560)-Net in Base 49 — Upper bound on s

There is no (15, 43, 19561)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 4 770393 822618 453931 152599 007604 722171 785471 558264 109670 813492 924282 337953 > 49⁴³ [i]