MinT - Best Known (39, 39+123, s)-Nets in Base 4

Best Known (39, 39+123, s)-Nets in Base 4

(39, 39+123, 56)-Net over F₄ — Constructive and digital

Digital (39, 162, 56)-net over F₄, using

t-expansion [i] based on digital (33, 162, 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, 39+123, 66)-Net over F₄ — Digital

Digital (39, 162, 66)-net over F₄, using

t-expansion [i] based on digital (37, 162, 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, 39+123, 165)-Net over F₄ — Upper bound on s (digital)

There is no digital (39, 162, 166)-net over F₄, because

3 times m-reduction [i] would yield digital (39, 159, 166)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁹, 166, F₄, 120) (dual of [166, 7, 121]-code), but
  - residual code [i] would yield linear OA(4³⁹, 45, F₄, 30) (dual of [45, 6, 31]-code), but
    - â€œDaâ€ bound on codes from Brouwerâ€™s database [i]

(39, 39+123, 257)-Net in Base 4 — Upper bound on s

There is no (39, 162, 258)-net in base 4, because

1 times m-reduction [i] would yield (39, 161, 258)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 479533 511384 400588 842028 157356 507545 105951 608619 072910 407185 947025 200875 464085 830591 446793 405120 > 4¹⁶¹ [i]