MinT - Best Known (149−107, 149, s)-Nets in Base 4

Best Known (149−107, 149, s)-Nets in Base 4

(149−107, 149, 56)-Net over F₄ — Constructive and digital

Digital (42, 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]

(149−107, 149, 75)-Net over F₄ — Digital

Digital (42, 149, 75)-net over F₄, using

t-expansion [i] based on digital (40, 149, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

(149−107, 149, 276)-Net over F₄ — Upper bound on s (digital)

There is no digital (42, 149, 277)-net over F₄, because

3 times m-reduction [i] would yield digital (42, 146, 277)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁶, 277, F₄, 104) (dual of [277, 131, 105]-code), but
  - residual code [i] would yield OA(4⁴², 172, S₄, 26), but
    - the linear programming bound shows that MÂ â‰¥ 34 487674 602431 075505 978819 529968 455340 679044 792320 / 1 686703 652728 844301 656717 > 4⁴² [i]

(149−107, 149, 287)-Net in Base 4 — Upper bound on s

There is no (42, 149, 288)-net in base 4, because

1 times m-reduction [i] would yield (42, 148, 288)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 128680 410551 731603 292114 670156 737881 407326 842181 248968 250962 136084 863168 532450 007171 423800 > 4¹⁴⁸ [i]