MinT - Best Known (149−110, 149, s)-Nets in Base 4

Best Known (149−110, 149, s)-Nets in Base 4

(149−110, 149, 56)-Net over F₄ — Constructive and digital

Digital (39, 149, 56)-net over F₄, using

t-expansion [i] based on digital (33, 149, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(149−110, 149, 66)-Net over F₄ — Digital

Digital (39, 149, 66)-net over F₄, using

t-expansion [i] based on digital (37, 149, 66)-net over F₄, using
- net from sequence [i] based on digital (37, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
    - Drinfeld modules of rank 1 [i]

(149−110, 149, 209)-Net over F₄ — Upper bound on s (digital)

There is no digital (39, 149, 210)-net over F₄, because

2 times m-reduction [i] would yield digital (39, 147, 210)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁷, 210, F₄, 108) (dual of [210, 63, 109]-code), but
  - residual code [i] would yield OA(4³⁹, 101, S₄, 27), but
    - the linear programming bound shows that MÂ â‰¥ 2142 295795 661249 783425 772725 003117 015606 758101 383863 336960 / 7086 731333 443386 538541 653392 431983 > 4³⁹ [i]

(149−110, 149, 261)-Net in Base 4 — Upper bound on s

There is no (39, 149, 262)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 588453 367794 427931 967927 118520 584917 325589 673281 008024 049782 609842 168693 113521 845676 213120 > 4¹⁴⁹ [i]