MinT - Best Known (231−188, 231, s)-Nets in Base 4

Best Known (231−188, 231, s)-Nets in Base 4

(231−188, 231, 56)-Net over F₄ — Constructive and digital

Digital (43, 231, 56)-net over F₄, using

t-expansion [i] based on digital (33, 231, 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]

(231−188, 231, 75)-Net over F₄ — Digital

Digital (43, 231, 75)-net over F₄, using

t-expansion [i] based on digital (40, 231, 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]

(231−188, 231, 180)-Net over F₄ — Upper bound on s (digital)

There is no digital (43, 231, 181)-net over F₄, because

56 times m-reduction [i] would yield digital (43, 175, 181)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁷⁵, 181, F₄, 132) (dual of [181, 6, 133]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁷⁴, 178, F₄, 132) (dual of [178, 4, 133]-code), but
      - Griesmer bound [i]
    2. linear OA(4⁶, 181, F₄, 3) (dual of [181, 175, 4]-code or 181-cap in PG(5,4)), but
      - discarding factors / shortening the dual code would yield linear OA(4⁶, 160, F₄, 3) (dual of [160, 154, 4]-code or 160-cap in PG(5,4)), but
        1 times Hill recurrence [i] would yield linear OA(4⁵, 42, F₄, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,4)), but
        41 is the largest size of a cap in PG(4,Â 4) [i]

(231−188, 231, 186)-Net in Base 4 — Upper bound on s

There is no (43, 231, 187)-net in base 4, because

48 times m-reduction [i] would yield (43, 183, 187)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁸³, 187, S₄, 140), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 9619 630419 041620 901435 312524 449124 464130 795720 328478 190417 063819 395928 166869 436184 427311 097384 012607 618805 661696 / 47 > 4¹⁸³ [i]