MinT - Best Known (198−143, 198, s)-Nets in Base 4

Best Known (198−143, 198, s)-Nets in Base 4

(198−143, 198, 66)-Net over F₄ — Constructive and digital

Digital (55, 198, 66)-net over F₄, using

t-expansion [i] based on digital (49, 198, 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]

(198−143, 198, 91)-Net over F₄ — Digital

Digital (55, 198, 91)-net over F₄, using

t-expansion [i] based on digital (50, 198, 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]

(198−143, 198, 326)-Net over F₄ — Upper bound on s (digital)

There is no digital (55, 198, 327)-net over F₄, because

3 times m-reduction [i] would yield digital (55, 195, 327)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹⁵, 327, F₄, 140) (dual of [327, 132, 141]-code), but
  - residual code [i] would yield OA(4⁵⁵, 186, S₄, 35), but
    - the linear programming bound shows that MÂ â‰¥ 2227 845699 810146 058122 102603 602007 534656 013828 852609 458125 647380 480000 / 1 700436 329479 757404 993871 394563 863537 > 4⁵⁵ [i]

(198−143, 198, 369)-Net in Base 4 — Upper bound on s

There is no (55, 198, 370)-net in base 4, because

1 times m-reduction [i] would yield (55, 197, 370)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 41547 235088 435879 246952 449104 010012 301225 940288 480864 162803 131985 663424 824946 459069 236334 121329 557382 852610 985919 332904 > 4¹⁹⁷ [i]