MinT - Best Known (227−168, 227, s)-Nets in Base 4

Best Known (227−168, 227, s)-Nets in Base 4

(227−168, 227, 66)-Net over F₄ — Constructive and digital

Digital (59, 227, 66)-net over F₄, using

t-expansion [i] based on digital (49, 227, 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]

(227−168, 227, 91)-Net over F₄ — Digital

Digital (59, 227, 91)-net over F₄, using

t-expansion [i] based on digital (50, 227, 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]

(227−168, 227, 271)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 227, 272)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²⁷, 272, F₄, 168) (dual of [272, 45, 169]-code), but
- residual code [i] would yield OA(4⁵⁹, 103, S₄, 42), but
  - the linear programming bound shows that MÂ â‰¥ 20 209278 630007 954817 311657 592716 406577 471235 077989 401051 636395 089582 131646 221319 183990 947510 747136 / 59 997456 042944 145499 206694 218235 767252 553003 337898 939171 700875 > 4⁵⁹ [i]

(227−168, 227, 387)-Net in Base 4 — Upper bound on s

There is no (59, 227, 388)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53621 962404 101386 076641 493935 572266 036890 494645 483584 994796 576217 342492 146450 931173 645958 954763 247420 204464 551222 820120 759616 759111 829920 > 4²²⁷ [i]