MinT - Best Known (232−174, 232, s)-Nets in Base 4

Best Known (232−174, 232, s)-Nets in Base 4

(232−174, 232, 66)-Net over F₄ — Constructive and digital

Digital (58, 232, 66)-net over F₄, using

t-expansion [i] based on digital (49, 232, 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]

(232−174, 232, 91)-Net over F₄ — Digital

Digital (58, 232, 91)-net over F₄, using

t-expansion [i] based on digital (50, 232, 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]

(232−174, 232, 245)-Net over F₄ — Upper bound on s (digital)

There is no digital (58, 232, 246)-net over F₄, because

2 times m-reduction [i] would yield digital (58, 230, 246)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁰, 246, F₄, 172) (dual of [246, 16, 173]-code), but
  - residual code [i] would yield OA(4⁵⁸, 73, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 2189 401164 029144 651258 105721 344079 901877 075968 / 23088 690625 > 4⁵⁸ [i]

(232−174, 232, 378)-Net in Base 4 — Upper bound on s

There is no (58, 232, 379)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 56 925513 923223 126593 268121 974210 785770 174844 826807 043374 088759 099214 603954 841217 886457 919915 672342 622728 005053 329172 746378 549926 032875 094560 > 4²³² [i]