MinT - Best Known (56, 196, s)-Nets in Base 4

Best Known (56, 196, s)-Nets in Base 4

(56, 196, 66)-Net over F₄ — Constructive and digital

Digital (56, 196, 66)-net over F₄, using

t-expansion [i] based on digital (49, 196, 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]

(56, 196, 91)-Net over F₄ — Digital

Digital (56, 196, 91)-net over F₄, using

t-expansion [i] based on digital (50, 196, 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]

(56, 196, 344)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 196, 345)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁹⁶, 345, F₄, 140) (dual of [345, 149, 141]-code), but
- residual code [i] would yield OA(4⁵⁶, 204, S₄, 35), but
  - the linear programming bound shows that MÂ â‰¥ 78295 866271 954113 495370 355913 581218 786159 455204 222712 111589 032673 922357 657600 / 14 695871 573792 476851 963908 736113 155530 258961 > 4⁵⁶ [i]

(56, 196, 379)-Net in Base 4 — Upper bound on s

There is no (56, 196, 380)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10481 959370 025305 887807 265985 761681 185351 932266 600265 644781 141318 720325 939150 779771 782977 811875 403724 363470 229278 621326 > 4¹⁹⁶ [i]