MinT - Best Known (11, 30, s)-Nets in Base 49

Best Known (11, 30, s)-Nets in Base 49

(11, 30, 102)-Net over F₄₉ — Constructive and digital

Digital (11, 30, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 20, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(11, 30, 128)-Net over F₄₉ — Digital

Digital (11, 30, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49³⁰, 128, F₄₉, 3, 19) (dual of [(128, 3), 354, 20]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹⁰, 64, F₄₉, 3, 9) (dual of [(64, 3), 182, 10]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,182P) [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(49²⁰, 64, F₄₉, 3, 19) (dual of [(64, 3), 172, 20]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,172P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(11, 30, 24133)-Net in Base 49 — Upper bound on s

There is no (11, 30, 24134)-net in base 49, because

1 times m-reduction [i] would yield (11, 29, 24134)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 10 369313 006769 607331 483739 649988 931085 560277 772833 > 49²⁹ [i]