MinT - Best Known (13, 13+22, s)-Nets in Base 49

Best Known (13, 13+22, s)-Nets in Base 49

(13, 13+22, 102)-Net over F₄₉ — Constructive and digital

Digital (13, 35, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 23, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(13, 13+22, 128)-Net over F₄₉ — Digital

Digital (13, 35, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49³⁵, 128, F₄₉, 2, 22) (dual of [(128, 2), 221, 23]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹², 64, F₄₉, 2, 11) (dual of [(64, 2), 116, 12]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,116P) [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]
  2. linear OOA(49²³, 64, F₄₉, 2, 22) (dual of [(64, 2), 105, 23]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,105P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(13, 13+22, 24410)-Net in Base 49 — Upper bound on s

There is no (13, 35, 24411)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 143536 503005 300218 430345 837333 946101 813734 688429 987356 791217 > 49³⁵ [i]