MinT - Best Known (39, 144, s)-Nets in Base 4

Best Known (39, 144, s)-Nets in Base 4

(39, 144, 56)-Net over F₄ — Constructive and digital

Digital (39, 144, 56)-net over F₄, using

t-expansion [i] based on digital (33, 144, 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]

(39, 144, 66)-Net over F₄ — Digital

Digital (39, 144, 66)-net over F₄, using

t-expansion [i] based on digital (37, 144, 66)-net over F₄, using
- net from sequence [i] based on digital (37, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
    - Drinfeld modules of rank 1 [i]

(39, 144, 221)-Net over F₄ — Upper bound on s (digital)

There is no digital (39, 144, 222)-net over F₄, because

1 times m-reduction [i] would yield digital (39, 143, 222)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴³, 222, F₄, 104) (dual of [222, 79, 105]-code), but
  - residual code [i] would yield OA(4³⁹, 117, S₄, 26), but
    - the linear programming bound shows that MÂ â‰¥ 363759 748775 180038 987523 636245 734966 329724 329420 390400 / 1 130801 960300 911622 852867 106887 > 4³⁹ [i]

(39, 144, 264)-Net in Base 4 — Upper bound on s

There is no (39, 144, 265)-net in base 4, because

1 times m-reduction [i] would yield (39, 143, 265)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 137 536111 728799 851420 473586 199845 121525 430043 045604 711635 937955 631321 367008 139698 739936 > 4¹⁴³ [i]