MinT - Best Known (27−17, 27, s)-Nets in Base 49

Best Known (27−17, 27, s)-Nets in Base 49

(27−17, 27, 102)-Net over F₄₉ — Constructive and digital

Digital (10, 27, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 18, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(27−17, 27, 128)-Net over F₄₉ — Digital

Digital (10, 27, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49²⁷, 128, F₄₉, 3, 17) (dual of [(128, 3), 357, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49⁹, 64, F₄₉, 3, 8) (dual of [(64, 3), 183, 9]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,183P) [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(49¹⁸, 64, F₄₉, 3, 17) (dual of [(64, 3), 174, 18]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,174P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(27−17, 27, 24407)-Net in Base 49 — Upper bound on s

There is no (10, 27, 24408)-net in base 49, because

1 times m-reduction [i] would yield (10, 26, 24408)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 88 143198 552520 565329 433477 864925 644556 063745 > 49²⁶ [i]