MinT - Best Known (192−137, 192, s)-Nets in Base 4

Best Known (192−137, 192, s)-Nets in Base 4

(192−137, 192, 66)-Net over F₄ — Constructive and digital

Digital (55, 192, 66)-net over F₄, using

t-expansion [i] based on digital (49, 192, 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]

(192−137, 192, 91)-Net over F₄ — Digital

Digital (55, 192, 91)-net over F₄, using

t-expansion [i] based on digital (50, 192, 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]

(192−137, 192, 347)-Net over F₄ — Upper bound on s (digital)

There is no digital (55, 192, 348)-net over F₄, because

1 times m-reduction [i] would yield digital (55, 191, 348)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹¹, 348, F₄, 136) (dual of [348, 157, 137]-code), but
  - residual code [i] would yield OA(4⁵⁵, 211, S₄, 34), but
    - the linear programming bound shows that MÂ â‰¥ 10743 840493 216536 083692 298711 093271 763797 668298 834882 461033 299968 000000 / 7 972480 994288 747790 242491 803609 004511 > 4⁵⁵ [i]

(192−137, 192, 374)-Net in Base 4 — Upper bound on s

There is no (55, 192, 375)-net in base 4, because

1 times m-reduction [i] would yield (55, 191, 375)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 588017 595111 587428 994942 912551 746038 234823 943404 647024 492277 767273 709516 834819 737904 740771 317535 084846 268121 165221 > 4¹⁹¹ [i]