MinT - Best Known (251−209, 251, s)-Nets in Base 4

Best Known (251−209, 251, s)-Nets in Base 4

(251−209, 251, 56)-Net over F₄ — Constructive and digital

Digital (42, 251, 56)-net over F₄, using

t-expansion [i] based on digital (33, 251, 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]

(251−209, 251, 75)-Net over F₄ — Digital

Digital (42, 251, 75)-net over F₄, using

t-expansion [i] based on digital (40, 251, 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]

(251−209, 251, 175)-Net over F₄ — Upper bound on s (digital)

There is no digital (42, 251, 176)-net over F₄, because

81 times m-reduction [i] would yield digital (42, 170, 176)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁷⁰, 176, F₄, 128) (dual of [176, 6, 129]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁶⁹, 173, F₄, 128) (dual of [173, 4, 129]-code), but
      - Griesmer bound [i]
    2. linear OA(4⁶, 176, F₄, 3) (dual of [176, 170, 4]-code or 176-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]

(251−209, 251, 182)-Net in Base 4 — Upper bound on s

There is no (42, 251, 183)-net in base 4, because

72 times m-reduction [i] would yield (42, 179, 183)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁷⁹, 183, S₄, 137), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 18 788340 662190 665823 115844 774314 696219 005460 391266 558965 658327 772257 672200 916867 547709 591987 078149 624255 479808 / 23 > 4¹⁷⁹ [i]