MinT - Best Known (58, 58+172, s)-Nets in Base 4

Best Known (58, 58+172, s)-Nets in Base 4

(58, 58+172, 66)-Net over F₄ — Constructive and digital

Digital (58, 230, 66)-net over F₄, using

t-expansion [i] based on digital (49, 230, 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]

(58, 58+172, 91)-Net over F₄ — Digital

Digital (58, 230, 91)-net over F₄, using

t-expansion [i] based on digital (50, 230, 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]

(58, 58+172, 245)-Net over F₄ — Upper bound on s (digital)

There is no digital (58, 230, 246)-net over F₄, because

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]

(58, 58+172, 378)-Net in Base 4 — Upper bound on s

There is no (58, 230, 379)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 197562 441530 351802 369485 869749 747330 447094 499799 289055 116372 524831 752914 176615 311676 097445 721190 096156 491158 214989 562992 596911 694973 543782 > 4²³⁰ [i]