MinT - Best Known (28−17, 28, s)-Nets in Base 49

Best Known (28−17, 28, s)-Nets in Base 49

(28−17, 28, 103)-Net over F₄₉ — Constructive and digital

Digital (11, 28, 103)-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 (2, 19, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(28−17, 28, 142)-Net over F₄₉ — Digital

Digital (11, 28, 142)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49²⁸, 142, F₄₉, 17) (dual of [142, 114, 18]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(49⁹, 64, F₄₉, 8) (dual of [64, 55, 9]-code), using
    - extended algebraic-geometric code AG_e(F,55P) [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 OA(49¹⁹, 78, F₄₉, 17) (dual of [78, 59, 18]-code), using
    - extended algebraic-geometric code AG_e(F,60P) [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]

(28−17, 28, 39702)-Net in Base 49 — Upper bound on s

There is no (11, 28, 39703)-net in base 49, because

1 times m-reduction [i] would yield (11, 27, 39703)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 4318 339757 364364 833845 587864 353566 329469 948545 > 49²⁷ [i]