MinT - Best Known (42, 42+128, s)-Nets in Base 4

Best Known (42, 42+128, s)-Nets in Base 4

(42, 42+128, 56)-Net over F₄ — Constructive and digital

Digital (42, 170, 56)-net over F₄, using

t-expansion [i] based on digital (33, 170, 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]

(42, 42+128, 75)-Net over F₄ — Digital

Digital (42, 170, 75)-net over F₄, using

t-expansion [i] based on digital (40, 170, 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]

(42, 42+128, 175)-Net over F₄ — Upper bound on s (digital)

There is no digital (42, 170, 176)-net over F₄, because

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]

(42, 42+128, 277)-Net in Base 4 — Upper bound on s

There is no (42, 170, 278)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 715055 888763 231855 791895 415647 368630 352960 546735 165398 878741 928641 382627 775300 379462 019344 575877 672940 > 4¹⁷⁰ [i]