MinT - Best Known (10, 10+18, s)-Nets in Base 49

Best Known (10, 10+18, s)-Nets in Base 49

(10, 10+18, 101)-Net over F₄₉ — Constructive and digital

Digital (10, 28, 101)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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, 19, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(10, 10+18, 114)-Net over F₄₉ — Digital

Digital (10, 28, 114)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49²⁸, 114, F₄₉, 3, 18) (dual of [(114, 3), 314, 19]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49⁹, 50, F₄₉, 3, 9) (dual of [(50, 3), 141, 10]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(3;141,49) [i]
  2. linear OOA(49¹⁹, 64, F₄₉, 3, 18) (dual of [(64, 3), 173, 19]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,173P) [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]

(10, 10+18, 15659)-Net in Base 49 — Upper bound on s

There is no (10, 28, 15660)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 211630 810445 323898 559230 860101 388130 808916 685377 > 49²⁸ [i]