MinT - Best Known (127−78, 127, s)-Nets in Base 9

Best Known (127−78, 127, s)-Nets in Base 9

(127−78, 127, 81)-Net over F₉ — Constructive and digital

Digital (49, 127, 81)-net over F₉, using

t-expansion [i] based on digital (32, 127, 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]

(127−78, 127, 168)-Net over F₉ — Digital

Digital (49, 127, 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]

(127−78, 127, 2440)-Net in Base 9 — Upper bound on s

There is no (49, 127, 2441)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15 504588 456170 120738 410738 933087 251488 181304 046027 681192 764317 459395 331351 755375 665065 101625 168101 803816 455339 474833 902905 > 9¹²⁷ [i]