MinT - Best Known (243−180, 243, s)-Nets in Base 4

Best Known (243−180, 243, s)-Nets in Base 4

(243−180, 243, 66)-Net over F₄ — Constructive and digital

Digital (63, 243, 66)-net over F₄, using

t-expansion [i] based on digital (49, 243, 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]

(243−180, 243, 99)-Net over F₄ — Digital

Digital (63, 243, 99)-net over F₄, using

t-expansion [i] based on digital (61, 243, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(243−180, 243, 283)-Net over F₄ — Upper bound on s (digital)

There is no digital (63, 243, 284)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴³, 284, F₄, 180) (dual of [284, 41, 181]-code), but
- residual code [i] would yield OA(4⁶³, 103, S₄, 45), but
  - the linear programming bound shows that MÂ â‰¥ 30 963907 201042 572946 047598 093595 047914 220745 773823 416057 277667 026891 964416 / 314660 526989 952486 919791 531363 400925 > 4⁶³ [i]

(243−180, 243, 412)-Net in Base 4 — Upper bound on s

There is no (63, 243, 413)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 229 193582 373861 400840 480596 167606 584160 649541 229711 105951 035574 612258 206180 543018 978852 005548 373446 094277 710648 614825 068258 468411 398822 312617 925700 > 4²⁴³ [i]