MinT - Best Known (42, 42+113, s)-Nets in Base 4

Best Known (42, 42+113, s)-Nets in Base 4

(42, 42+113, 56)-Net over F₄ — Constructive and digital

Digital (42, 155, 56)-net over F₄, using

t-expansion [i] based on digital (33, 155, 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]

(42, 42+113, 75)-Net over F₄ — Digital

Digital (42, 155, 75)-net over F₄, using

t-expansion [i] based on digital (40, 155, 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]

(42, 42+113, 235)-Net over F₄ — Upper bound on s (digital)

There is no digital (42, 155, 236)-net over F₄, because

1 times m-reduction [i] would yield digital (42, 154, 236)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁴, 236, F₄, 112) (dual of [236, 82, 113]-code), but
  - residual code [i] would yield OA(4⁴², 123, S₄, 28), but
    - the linear programming bound shows that MÂ â‰¥ 420413 078661 724564 093040 229341 896213 924456 713137 684480 / 21073 249104 767480 966531 060729 > 4⁴² [i]

(42, 42+113, 283)-Net in Base 4 — Upper bound on s

There is no (42, 155, 284)-net in base 4, because

1 times m-reduction [i] would yield (42, 154, 284)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 555 487707 517566 069263 304071 074696 710040 059990 016155 447561 669062 352312 187167 952119 820314 917320 > 4¹⁵⁴ [i]