MinT - Best Known (40, 40+116, s)-Nets in Base 4

Best Known (40, 40+116, s)-Nets in Base 4

(40, 40+116, 56)-Net over F₄ — Constructive and digital

Digital (40, 156, 56)-net over F₄, using

t-expansion [i] based on digital (33, 156, 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, 40+116, 75)-Net over F₄ — Digital

Digital (40, 156, 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, 40+116, 189)-Net over F₄ — Upper bound on s (digital)

There is no digital (40, 156, 190)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁶, 190, F₄, 116) (dual of [190, 34, 117]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(4¹⁵⁵, 170, F₄, 116) (dual of [170, 15, 117]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(4¹⁵⁴, 162, F₄, 116) (dual of [162, 8, 117]-code), but
        constructionÂ Y1 [i] would yield
        linear OA(4¹⁵³, 158, F₄, 116) (dual of [158, 5, 117]-code), but
        residual code [i] would yield linear OA(4³⁷, 41, F₄, 29) (dual of [41, 4, 30]-code), but
        1 times truncation [i] would yield linear OA(4³⁶, 40, F₄, 28) (dual of [40, 4, 29]-code), but
        residual code [i] would yield linear OA(4⁸, 11, F₄, 7) (dual of [11, 3, 8]-code), but
        improvement by Maruta on the Griesmer bound [i]
        OA(4⁸, 162, S₄, 4), but
        discarding factors would yield OA(4⁸, 121, S₄, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 65704 > 4⁸ [i]
      2. OA(4¹⁵, 170, S₄, 8), but
        discarding factors would yield OA(4¹⁵, 135, S₄, 8), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1082 768311 > 4¹⁵ [i]
  2. OA(4³⁴, 190, S₄, 20), but
    - discarding factors would yield OA(4³⁴, 173, S₄, 20), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 305 996090 287239 486286 > 4³⁴ [i]

(40, 40+116, 266)-Net in Base 4 — Upper bound on s

There is no (40, 156, 267)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9592 192330 021642 682090 208449 965863 531809 813064 796898 198517 222430 004502 281809 474998 623664 801904 > 4¹⁵⁶ [i]