MinT - Best Known (40, 162, s)-Nets in Base 4

Best Known (40, 162, s)-Nets in Base 4

(40, 162, 56)-Net over F₄ — Constructive and digital

Digital (40, 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]

(40, 162, 75)-Net over F₄ — Digital

Digital (40, 162, 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, 162, 171)-Net over F₄ — Upper bound on s (digital)

There is no digital (40, 162, 172)-net over F₄, because

2 times m-reduction [i] would yield digital (40, 160, 172)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁶⁰, 172, F₄, 120) (dual of [172, 12, 121]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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]
    2. OA(4¹², 172, S₄, 6), but
      - discarding factors would yield OA(4¹², 156, S₄, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 16 866019 > 4¹² [i]

(40, 162, 264)-Net in Base 4 — Upper bound on s

There is no (40, 162, 265)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38 223091 276387 847682 895389 943113 593075 909306 568398 137275 836178 389674 266040 901327 165948 187140 167168 > 4¹⁶² [i]