MinT - Best Known (216−156, 216, s)-Nets in Base 4

Best Known (216−156, 216, s)-Nets in Base 4

(216−156, 216, 66)-Net over F₄ — Constructive and digital

Digital (60, 216, 66)-net over F₄, using

t-expansion [i] based on digital (49, 216, 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]

(216−156, 216, 91)-Net over F₄ — Digital

Digital (60, 216, 91)-net over F₄, using

t-expansion [i] based on digital (50, 216, 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]

(216−156, 216, 336)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 216, 337)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²¹⁶, 337, F₄, 156) (dual of [337, 121, 157]-code), but
- residual code [i] would yield OA(4⁶⁰, 180, S₄, 39), but
  - the linear programming bound shows that MÂ â‰¥ 64 592000 643692 089079 054839 691926 765927 402602 779909 340826 367902 235627 537241 992340 655885 516800 / 46 756581 550576 177840 289976 743881 661381 961798 582533 083517 > 4⁶⁰ [i]

(216−156, 216, 401)-Net in Base 4 — Upper bound on s

There is no (60, 216, 402)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12560 392340 511234 242407 813721 119642 870027 382959 509382 472615 321393 806219 509767 975313 450128 231809 229290 862033 640449 765335 401614 569744 > 4²¹⁶ [i]