MinT - Best Known (57, 57+65, s)-Nets in Base 4

Best Known (57, 57+65, s)-Nets in Base 4

(57, 57+65, 66)-Net over F₄ — Constructive and digital

Digital (57, 122, 66)-net over F₄, using

t-expansion [i] based on digital (49, 122, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(57, 57+65, 91)-Net over F₄ — Digital

Digital (57, 122, 91)-net over F₄, using

t-expansion [i] based on digital (50, 122, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(57, 57+65, 780)-Net in Base 4 — Upper bound on s

There is no (57, 122, 781)-net in base 4, because

1 times m-reduction [i] would yield (57, 121, 781)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 328429 069201 418342 034196 453755 494978 880235 149745 794917 200184 294611 879944 > 4¹²¹ [i]