MinT - Best Known (33−21, 33, s)-Nets in Base 49

Best Known (33−21, 33, s)-Nets in Base 49

(33−21, 33, 102)-Net over F₄₉ — Constructive and digital

Digital (12, 33, 102)-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 (1, 22, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(33−21, 33, 128)-Net over F₄₉ — Digital

Digital (12, 33, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49³³, 128, F₄₉, 3, 21) (dual of [(128, 3), 351, 22]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹¹, 64, F₄₉, 3, 10) (dual of [(64, 3), 181, 11]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,181P) [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(49²², 64, F₄₉, 3, 21) (dual of [(64, 3), 170, 22]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,170P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(33−21, 33, 24170)-Net in Base 49 — Upper bound on s

There is no (12, 33, 24171)-net in base 49, because

1 times m-reduction [i] would yield (12, 32, 24171)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 1 220229 100866 061182 359948 127542 326759 457472 108408 865441 > 49³² [i]