MinT - Best Known (41, 162, s)-Nets in Base 4

Best Known (41, 162, s)-Nets in Base 4

(41, 162, 56)-Net over F₄ — Constructive and digital

Digital (41, 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]

(41, 162, 75)-Net over F₄ — Digital

Digital (41, 162, 75)-net over F₄, using

t-expansion [i] based on 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]

(41, 162, 185)-Net over F₄ — Upper bound on s (digital)

There is no digital (41, 162, 186)-net over F₄, because

1 times m-reduction [i] would yield digital (41, 161, 186)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁶¹, 186, F₄, 120) (dual of [186, 25, 121]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁶⁰, 172, F₄, 120) (dual of [172, 12, 121]-code), but
      - constructionÂ Y1 [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]
        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]
    2. OA(4²⁵, 186, S₄, 14), but
      - discarding factors would yield OA(4²⁵, 162, S₄, 14), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1131 065428 484512 > 4²⁵ [i]

(41, 162, 272)-Net in Base 4 — Upper bound on s

There is no (41, 162, 273)-net in base 4, because

1 times m-reduction [i] would yield (41, 161, 273)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 972059 416038 090367 909591 332213 327328 350705 691596 195447 740617 211883 939692 608759 124084 549085 307456 > 4¹⁶¹ [i]