MinT - Best Known (40, 40+109, s)-Nets in Base 4

Best Known (40, 40+109, s)-Nets in Base 4

(40, 40+109, 56)-Net over F₄ — Constructive and digital

Digital (40, 149, 56)-net over F₄, using

t-expansion [i] based on digital (33, 149, 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+109, 75)-Net over F₄ — Digital

Digital (40, 149, 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+109, 223)-Net over F₄ — Upper bound on s (digital)

There is no digital (40, 149, 224)-net over F₄, because

1 times m-reduction [i] would yield digital (40, 148, 224)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁸, 224, F₄, 108) (dual of [224, 76, 109]-code), but
  - residual code [i] would yield OA(4⁴⁰, 115, S₄, 27), but
    - the linear programming bound shows that MÂ â‰¥ 929 557766 490609 787836 487564 000967 310274 723840 000000 / 708 347236 523196 849504 526399 > 4⁴⁰ [i]

(40, 40+109, 270)-Net in Base 4 — Upper bound on s

There is no (40, 149, 271)-net in base 4, because

1 times m-reduction [i] would yield (40, 148, 271)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 151053 313883 306955 371053 792811 652335 632703 321848 507796 356777 108756 226660 360174 056203 243856 > 4¹⁴⁸ [i]