MinT - Best Known (243−182, 243, s)-Nets in Base 4

Best Known (243−182, 243, s)-Nets in Base 4

(243−182, 243, 66)-Net over F₄ — Constructive and digital

Digital (61, 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]

(243−182, 243, 99)-Net over F₄ — Digital

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]

(243−182, 243, 257)-Net over F₄ — Upper bound on s (digital)

There is no digital (61, 243, 258)-net over F₄, because

2 times m-reduction [i] would yield digital (61, 241, 258)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴¹, 258, F₄, 180) (dual of [258, 17, 181]-code), but
  - residual code [i] would yield OA(4⁶¹, 77, S₄, 45), but
    - the linear programming bound shows that MÂ â‰¥ 87771 116150 262149 392012 316054 470316 656445 882368 / 16226 709967 > 4⁶¹ [i]

(243−182, 243, 397)-Net in Base 4 — Upper bound on s

There is no (61, 243, 398)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 234 376895 967291 759282 364634 638558 635549 796130 572201 403664 923936 824423 586835 166139 420474 565691 066882 753691 204732 052295 541368 792668 193964 770527 297040 > 4²⁴³ [i]