MinT - Best Known (226−188, 226, s)-Nets in Base 4

Best Known (226−188, 226, s)-Nets in Base 4

(226−188, 226, 56)-Net over F₄ — Constructive and digital

Digital (38, 226, 56)-net over F₄, using

t-expansion [i] based on digital (33, 226, 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]

(226−188, 226, 66)-Net over F₄ — Digital

Digital (38, 226, 66)-net over F₄, using

t-expansion [i] based on digital (37, 226, 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]

(226−188, 226, 161)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 226, 162)-net over F₄, because

72 times m-reduction [i] would yield digital (38, 154, 162)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁴, 162, F₄, 116) (dual of [162, 8, 117]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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]
    2. 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]

(226−188, 226, 165)-Net in Base 4 — Upper bound on s

There is no (38, 226, 166)-net in base 4, because

64 times m-reduction [i] would yield (38, 162, 166)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁶², 166, S₄, 124), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 4374 501449 566023 848745 004454 235242 730706 338861 786424 872851 541212 819905 998398 751846 447026 354046 107648 / 125 > 4¹⁶² [i]