MinT - Best Known (40, 40+103, s)-Nets in Base 4

Best Known (40, 40+103, s)-Nets in Base 4

(40, 40+103, 56)-Net over F₄ — Constructive and digital

Digital (40, 143, 56)-net over F₄, using

t-expansion [i] based on digital (33, 143, 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+103, 75)-Net over F₄ — Digital

Digital (40, 143, 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+103, 264)-Net over F₄ — Upper bound on s (digital)

There is no digital (40, 143, 265)-net over F₄, because

3 times m-reduction [i] would yield digital (40, 140, 265)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁰, 265, F₄, 100) (dual of [265, 125, 101]-code), but
  - residual code [i] would yield OA(4⁴⁰, 164, S₄, 25), but
    - the linear programming bound shows that MÂ â‰¥ 14 570521 423098 291305 264511 747942 665498 471933 411328 / 11 371893 759058 304198 150255 > 4⁴⁰ [i]

(40, 40+103, 274)-Net in Base 4 — Upper bound on s

There is no (40, 143, 275)-net in base 4, because

1 times m-reduction [i] would yield (40, 142, 275)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35 978577 479243 463792 127598 964083 993742 210195 378222 980871 362829 737325 440363 798933 897056 > 4¹⁴² [i]