MinT - Best Known (175−132, 175, s)-Nets in Base 4

Best Known (175−132, 175, s)-Nets in Base 4

(175−132, 175, 56)-Net over F₄ — Constructive and digital

Digital (43, 175, 56)-net over F₄, using

t-expansion [i] based on digital (33, 175, 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]

(175−132, 175, 75)-Net over F₄ — Digital

Digital (43, 175, 75)-net over F₄, using

t-expansion [i] based on digital (40, 175, 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]

(175−132, 175, 180)-Net over F₄ — Upper bound on s (digital)

There is no digital (43, 175, 181)-net over F₄, because

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]

(175−132, 175, 283)-Net in Base 4 — Upper bound on s

There is no (43, 175, 284)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2750 332249 782135 626846 201554 521174 691996 640976 033876 040316 909384 996768 052476 585434 326089 787640 868551 596340 > 4¹⁷⁵ [i]