MinT - Best Known (57, 57+142, s)-Nets in Base 4

Best Known (57, 57+142, s)-Nets in Base 4

(57, 57+142, 66)-Net over F₄ — Constructive and digital

Digital (57, 199, 66)-net over F₄, using

t-expansion [i] based on digital (49, 199, 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+142, 91)-Net over F₄ — Digital

Digital (57, 199, 91)-net over F₄, using

t-expansion [i] based on digital (50, 199, 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+142, 366)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 199, 367)-net over F₄, because

2 times m-reduction [i] would yield digital (57, 197, 367)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹⁷, 367, F₄, 140) (dual of [367, 170, 141]-code), but
  - residual code [i] would yield OA(4⁵⁷, 226, S₄, 35), but
    - the linear programming bound shows that MÂ â‰¥ 206446 636819 354140 257573 963708 426022 937166 140581 505377 357019 798831 104000 / 9 666005 460414 349955 896849 894230 574741 > 4⁵⁷ [i]

(57, 57+142, 386)-Net in Base 4 — Upper bound on s

There is no (57, 199, 387)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 680753 621464 881775 770959 075518 372889 198722 024932 164469 987123 308256 123796 412757 731004 556800 106520 965850 842011 328258 738432 > 4¹⁹⁹ [i]