MinT - Best Known (156−115, 156, s)-Nets in Base 4

Best Known (156−115, 156, s)-Nets in Base 4

(156−115, 156, 56)-Net over F₄ — Constructive and digital

Digital (41, 156, 56)-net over F₄, using

t-expansion [i] based on digital (33, 156, 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]

(156−115, 156, 75)-Net over F₄ — Digital

Digital (41, 156, 75)-net over F₄, using

t-expansion [i] based on digital (40, 156, 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]

(156−115, 156, 223)-Net over F₄ — Upper bound on s (digital)

There is no digital (41, 156, 224)-net over F₄, because

3 times m-reduction [i] would yield digital (41, 153, 224)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵³, 224, F₄, 112) (dual of [224, 71, 113]-code), but
  - residual code [i] would yield OA(4⁴¹, 111, S₄, 28), but
    - the linear programming bound shows that MÂ â‰¥ 7709 702911 510783 693838 831699 200582 723072 520355 840000 / 1472 084210 434687 540905 417611 > 4⁴¹ [i]

(156−115, 156, 274)-Net in Base 4 — Upper bound on s

There is no (41, 156, 275)-net in base 4, because

1 times m-reduction [i] would yield (41, 155, 275)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2219 963707 114510 096840 677277 410595 954115 869252 056124 433044 090635 010868 092980 530460 965217 136576 > 4¹⁵⁵ [i]