MinT - Best Known (41, 41+107, s)-Nets in Base 4

Best Known (41, 41+107, s)-Nets in Base 4

(41, 41+107, 56)-Net over F₄ — Constructive and digital

Digital (41, 148, 56)-net over F₄, using

t-expansion [i] based on digital (33, 148, 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]

(41, 41+107, 75)-Net over F₄ — Digital

Digital (41, 148, 75)-net over F₄, using

t-expansion [i] based on digital (40, 148, 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]

(41, 41+107, 259)-Net over F₄ — Upper bound on s (digital)

There is no digital (41, 148, 260)-net over F₄, because

3 times m-reduction [i] would yield digital (41, 145, 260)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁵, 260, F₄, 104) (dual of [260, 115, 105]-code), but
  - residual code [i] would yield OA(4⁴¹, 155, S₄, 26), but
    - the linear programming bound shows that MÂ â‰¥ 552818 318951 874532 269123 698121 730294 556917 760000 / 109558 651860 901659 660161 > 4⁴¹ [i]

(41, 41+107, 279)-Net in Base 4 — Upper bound on s

There is no (41, 148, 280)-net in base 4, because

1 times m-reduction [i] would yield (41, 147, 280)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34709 963059 702462 778118 548066 973839 701738 706825 078098 153664 806385 551696 630484 285907 741400 > 4¹⁴⁷ [i]