MinT - Best Known (178−134, 178, s)-Nets in Base 4

Best Known (178−134, 178, s)-Nets in Base 4

(178−134, 178, 56)-Net over F₄ — Constructive and digital

Digital (44, 178, 56)-net over F₄, using

t-expansion [i] based on digital (33, 178, 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]

(178−134, 178, 75)-Net over F₄ — Digital

Digital (44, 178, 75)-net over F₄, using

t-expansion [i] based on digital (40, 178, 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]

(178−134, 178, 186)-Net over F₄ — Upper bound on s (digital)

There is no digital (44, 178, 187)-net over F₄, because

2 times m-reduction [i] would yield digital (44, 176, 187)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁷⁶, 187, F₄, 132) (dual of [187, 11, 133]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁷⁵, 181, F₄, 132) (dual of [181, 6, 133]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(4¹⁷⁴, 178, F₄, 132) (dual of [178, 4, 133]-code), but
        Griesmer bound [i]
        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]
    2. OA(4¹¹, 187, S₄, 6), but
      - discarding factors would yield OA(4¹¹, 99, S₄, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4 278880 > 4¹¹ [i]

(178−134, 178, 289)-Net in Base 4 — Upper bound on s

There is no (44, 178, 290)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 158512 971641 739192 852594 380407 663441 002855 997312 668878 944533 520379 113379 934123 397381 045594 758419 893541 975680 > 4¹⁷⁸ [i]