MinT - Best Known (16, 16+29, s)-Nets in Base 49

Best Known (16, 16+29, s)-Nets in Base 49

(16, 16+29, 102)-Net over F₄₉ — Constructive and digital

Digital (16, 45, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 30, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(16, 16+29, 128)-Net over F₄₉ — Digital

Digital (16, 45, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49⁴⁵, 128, F₄₉, 3, 29) (dual of [(128, 3), 339, 30]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹⁵, 64, F₄₉, 3, 14) (dual of [(64, 3), 177, 15]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,177P) [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(49³⁰, 64, F₄₉, 3, 29) (dual of [(64, 3), 162, 30]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,162P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(16, 16+29, 25831)-Net in Base 49 — Upper bound on s

There is no (16, 45, 25832)-net in base 49, because

1 times m-reduction [i] would yield (16, 44, 25832)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 233 765105 591145 369879 379590 797713 852886 066290 017564 145228 439885 338001 497345 > 49⁴⁴ [i]