MinT - Best Known (13, 34, s)-Nets in Base 49

Best Known (13, 34, s)-Nets in Base 49

(13, 34, 103)-Net over F₄₉ — Constructive and digital

Digital (13, 34, 103)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (2, 23, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(13, 34, 142)-Net over F₄₉ — Digital

Digital (13, 34, 142)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49³⁴, 142, F₄₉, 2, 21) (dual of [(142, 2), 250, 22]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹¹, 64, F₄₉, 2, 10) (dual of [(64, 2), 117, 11]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,117P) [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, 78, F₄₉, 2, 21) (dual of [(78, 2), 133, 22]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,134P) [i] based on function field F/F₄₉ with g(F) = 2 and N(F) ≥ 78, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(13, 34, 35671)-Net in Base 49 — Upper bound on s

There is no (13, 34, 35672)-net in base 49, because

1 times m-reduction [i] would yield (13, 33, 35672)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 59 771905 714769 656734 992458 869210 321473 916997 640142 455041 > 49³³ [i]