MinT - Best Known (17, 17+29, s)-Nets in Base 49

Best Known (17, 17+29, s)-Nets in Base 49

(17, 17+29, 103)-Net over F₄₉ — Constructive and digital

Digital (17, 46, 103)-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 (2, 31, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(17, 17+29, 142)-Net over F₄₉ — Digital

Digital (17, 46, 142)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49⁴⁶, 142, F₄₉, 2, 29) (dual of [(142, 2), 238, 30]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹⁵, 64, F₄₉, 2, 14) (dual of [(64, 2), 113, 15]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,113P) [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(49³¹, 78, F₄₉, 2, 29) (dual of [(78, 2), 125, 30]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,126P) [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]

(17, 17+29, 34111)-Net in Base 49 — Upper bound on s

There is no (17, 46, 34112)-net in base 49, because

1 times m-reduction [i] would yield (17, 45, 34112)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 11451 962228 541212 681986 862264 768619 630596 395610 396480 448232 998463 787591 510017 > 49⁴⁵ [i]