MinT - Best Known (66−44, 66, s)-Nets in Base 4

Best Known (66−44, 66, s)-Nets in Base 4

(66−44, 66, 34)-Net over F₄ — Constructive and digital

Digital (22, 66, 34)-net over F₄, using

t-expansion [i] based on digital (21, 66, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(66−44, 66, 44)-Net over F₄ — Digital

Digital (22, 66, 44)-net over F₄, using

t-expansion [i] based on digital (21, 66, 44)-net over F₄, using
- net from sequence [i] based on digital (21, 43)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 44, using
    - a curve from the manYPoints database [i]

(66−44, 66, 169)-Net in Base 4 — Upper bound on s

There is no (22, 66, 170)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶⁶, 170, S₄, 44), but
- the linear programming bound shows that MÂ â‰¥ 1976 506064 459033 075362 451713 190875 400705 748869 321511 254504 692102 419708 690447 989326 466803 280840 783010 320768 871264 419840 000000 / 348910 981362 978418 180472 516419 977086 112239 745527 698751 797516 935464 412701 432423 842977 > 4⁶⁶ [i]