MinT - Best Known (39−25, 39, s)-Nets in Base 49

Best Known (39−25, 39, s)-Nets in Base 49

(39−25, 39, 102)-Net over F₄₉ — Constructive and digital

Digital (14, 39, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 26, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(39−25, 39, 128)-Net over F₄₉ — Digital

Digital (14, 39, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49³⁹, 128, F₄₉, 3, 25) (dual of [(128, 3), 345, 26]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹³, 64, F₄₉, 3, 12) (dual of [(64, 3), 179, 13]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,179P) [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(49²⁶, 64, F₄₉, 3, 25) (dual of [(64, 3), 166, 26]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,166P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(39−25, 39, 24791)-Net in Base 49 — Upper bound on s

There is no (14, 39, 24792)-net in base 49, because

1 times m-reduction [i] would yield (14, 38, 24792)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 16885 158819 006622 780777 588043 664140 554930 712325 669375 030844 767745 > 49³⁸ [i]