MinT - Best Known (215−156, 215, s)-Nets in Base 4

Best Known (215−156, 215, s)-Nets in Base 4

(215−156, 215, 66)-Net over F₄ — Constructive and digital

Digital (59, 215, 66)-net over F₄, using

t-expansion [i] based on digital (49, 215, 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]

(215−156, 215, 91)-Net over F₄ — Digital

Digital (59, 215, 91)-net over F₄, using

t-expansion [i] based on digital (50, 215, 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]

(215−156, 215, 321)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 215, 322)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²¹⁵, 322, F₄, 156) (dual of [322, 107, 157]-code), but
- residual code [i] would yield OA(4⁵⁹, 165, S₄, 39), but
  - the linear programming bound shows that MÂ â‰¥ 3078 489369 750206 722287 954878 459506 651865 539185 921556 985227 028939 036190 618883 341680 640000 / 8667 454957 234129 743329 832681 955517 588182 622985 872027 > 4⁵⁹ [i]

(215−156, 215, 393)-Net in Base 4 — Upper bound on s

There is no (59, 215, 394)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3199 145667 163744 266053 409773 410821 462962 680398 356121 062067 500002 215587 403430 177843 495709 670719 671958 566370 180827 929344 602760 158528 > 4²¹⁵ [i]