MinT - Best Known (15, 42, s)-Nets in Base 49

Best Known (15, 42, s)-Nets in Base 49

(15, 42, 102)-Net over F₄₉ — Constructive and digital

Digital (15, 42, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 28, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(15, 42, 128)-Net over F₄₉ — Digital

Digital (15, 42, 128)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49⁴², 128, F₄₉, 3, 27) (dual of [(128, 3), 342, 28]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(49¹⁴, 64, F₄₉, 3, 13) (dual of [(64, 3), 178, 14]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,178P) [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(49²⁸, 64, F₄₉, 3, 27) (dual of [(64, 3), 164, 28]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,164P) [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64 (see above)

(15, 42, 25274)-Net in Base 49 — Upper bound on s

There is no (15, 42, 25275)-net in base 49, because

1 times m-reduction [i] would yield (15, 41, 25275)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 1987 199221 337017 969447 837050 616996 732414 621900 205332 467075 796980 569041 > 49⁴¹ [i]