MinT - Best Known (153−111, 153, s)-Nets in Base 4

Best Known (153−111, 153, s)-Nets in Base 4

(153−111, 153, 56)-Net over F₄ — Constructive and digital

Digital (42, 153, 56)-net over F₄, using

t-expansion [i] based on digital (33, 153, 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]

(153−111, 153, 75)-Net over F₄ — Digital

Digital (42, 153, 75)-net over F₄, using

t-expansion [i] based on digital (40, 153, 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]

(153−111, 153, 258)-Net over F₄ — Upper bound on s (digital)

There is no digital (42, 153, 259)-net over F₄, because

3 times m-reduction [i] would yield digital (42, 150, 259)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁰, 259, F₄, 108) (dual of [259, 109, 109]-code), but
  - residual code [i] would yield OA(4⁴², 150, S₄, 27), but
    - the linear programming bound shows that MÂ â‰¥ 6 907764 547941 000764 622618 877042 097977 075368 460288 000000 / 349879 061882 751493 871160 963067 > 4⁴² [i]

(153−111, 153, 284)-Net in Base 4 — Upper bound on s

There is no (42, 153, 285)-net in base 4, because

1 times m-reduction [i] would yield (42, 152, 285)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 32 971987 993485 492510 805914 838058 707086 199198 201801 917838 660276 847916 243528 227281 691688 043840 > 4¹⁵² [i]