MinT - Best Known (40, 40+115, s)-Nets in Base 4

Best Known (40, 40+115, s)-Nets in Base 4

(40, 40+115, 56)-Net over F₄ — Constructive and digital

Digital (40, 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]

(40, 40+115, 75)-Net over F₄ — Digital

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]

(40, 40+115, 208)-Net over F₄ — Upper bound on s (digital)

There is no digital (40, 155, 209)-net over F₄, because

3 times m-reduction [i] would yield digital (40, 152, 209)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵², 209, F₄, 112) (dual of [209, 57, 113]-code), but
  - residual code [i] would yield OA(4⁴⁰, 96, S₄, 28), but
    - the linear programming bound shows that MÂ â‰¥ 3 041221 052768 594873 805769 980859 249005 438281 138205 879215 063040 / 2 375396 786313 270370 198219 858144 081283 > 4⁴⁰ [i]

(40, 40+115, 267)-Net in Base 4 — Upper bound on s

There is no (40, 155, 268)-net in base 4, because

1 times m-reduction [i] would yield (40, 154, 268)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 622 215171 946356 122873 726566 736169 518302 883898 037692 188252 055751 571825 751214 369887 958144 211824 > 4¹⁵⁴ [i]