MinT - Best Known (39, 147, s)-Nets in Base 4

Best Known (39, 147, s)-Nets in Base 4

(39, 147, 56)-Net over F₄ — Constructive and digital

Digital (39, 147, 56)-net over F₄, using

t-expansion [i] based on digital (33, 147, 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, 147, 66)-Net over F₄ — Digital

Digital (39, 147, 66)-net over F₄, using

t-expansion [i] based on digital (37, 147, 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, 147, 209)-Net over F₄ — Upper bound on s (digital)

There is no digital (39, 147, 210)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁷, 210, F₄, 108) (dual of [210, 63, 109]-code), but
- residual code [i] would yield OA(4³⁹, 101, S₄, 27), but
  - the linear programming bound shows that MÂ â‰¥ 2142 295795 661249 783425 772725 003117 015606 758101 383863 336960 / 7086 731333 443386 538541 653392 431983 > 4³⁹ [i]

(39, 147, 262)-Net in Base 4 — Upper bound on s

There is no (39, 147, 263)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36995 318033 332479 320817 979541 787947 775957 178962 726916 799802 184788 268148 703640 221756 151696 > 4¹⁴⁷ [i]