MinT - Best Known (142−93, 142, s)-Nets in Base 9

Best Known (142−93, 142, s)-Nets in Base 9

(142−93, 142, 81)-Net over F₉ — Constructive and digital

Digital (49, 142, 81)-net over F₉, using

t-expansion [i] based on digital (32, 142, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(142−93, 142, 168)-Net over F₉ — Digital

Digital (49, 142, 168)-net over F₉, using

net from sequence [i] based on digital (49, 167)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 49 and N(F) ≥ 168, using
  - Drinfeld modules of rank 1 [i]

(142−93, 142, 1864)-Net in Base 9 — Upper bound on s

There is no (49, 142, 1865)-net in base 9, because

1 times m-reduction [i] would yield (49, 141, 1865)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 356 320971 728308 483335 378832 233175 777675 877697 647400 535096 665657 486944 311658 245849 105147 131560 449107 606405 861708 470003 882199 154766 398641 > 9¹⁴¹ [i]