MinT - Best Known (42, 42+125, s)-Nets in Base 4

Best Known (42, 42+125, s)-Nets in Base 4

(42, 42+125, 56)-Net over F₄ — Constructive and digital

Digital (42, 167, 56)-net over F₄, using

t-expansion [i] based on digital (33, 167, 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+125, 75)-Net over F₄ — Digital

Digital (42, 167, 75)-net over F₄, using

t-expansion [i] based on digital (40, 167, 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+125, 187)-Net over F₄ — Upper bound on s (digital)

There is no digital (42, 167, 188)-net over F₄, because

1 times m-reduction [i] would yield digital (42, 166, 188)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁶⁶, 188, F₄, 124) (dual of [188, 22, 125]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁶⁵, 176, F₄, 124) (dual of [176, 11, 125]-code), but
      - residual code [i] would yield linear OA(4⁴¹, 51, F₄, 31) (dual of [51, 10, 32]-code), but
        â€œGurâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(4²², 188, S₄, 12), but
      - discarding factors would yield OA(4²², 164, S₄, 12), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 18 187369 733464 > 4²² [i]

(42, 42+125, 278)-Net in Base 4 — Upper bound on s

There is no (42, 167, 279)-net in base 4, because

1 times m-reduction [i] would yield (42, 166, 279)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10313 105914 381279 196413 927581 878032 998449 899758 473432 670037 348366 390963 254028 060738 953023 776540 871200 > 4¹⁶⁶ [i]