MinT - Best Known (37−23, 37, s)-Nets in Base 49

Best Known (37−23, 37, s)-Nets in Base 49

(37−23, 37, 103)-Net over F₄₉ — Constructive and digital

Digital (14, 37, 103)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (2, 25, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(37−23, 37, 142)-Net over F₄₉ — Digital

Digital (14, 37, 142)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49³⁷, 142, F₄₉, 2, 23) (dual of [(142, 2), 247, 24]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹², 64, F₄₉, 2, 11) (dual of [(64, 2), 116, 12]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,116P) [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(49²⁵, 78, F₄₉, 2, 23) (dual of [(78, 2), 131, 24]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,132P) [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]

(37−23, 37, 34774)-Net in Base 49 — Upper bound on s

There is no (14, 37, 34775)-net in base 49, because

1 times m-reduction [i] would yield (14, 36, 34775)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 7 033228 064110 756764 355463 959357 632319 383988 505010 090957 993841 > 49³⁶ [i]