MinT - Best Known (157−120, 157, s)-Nets in Base 4

Best Known (157−120, 157, s)-Nets in Base 4

(157−120, 157, 56)-Net over F₄ — Constructive and digital

Digital (37, 157, 56)-net over F₄, using

t-expansion [i] based on digital (33, 157, 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]

(157−120, 157, 66)-Net over F₄ — Digital

Digital (37, 157, 66)-net over F₄, using

net from sequence [i] based on digital (37, 65)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
  - Drinfeld modules of rank 1 [i]

(157−120, 157, 156)-Net over F₄ — Upper bound on s (digital)

There is no digital (37, 157, 157)-net over F₄, because

8 times m-reduction [i] would yield digital (37, 149, 157)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁹, 157, F₄, 112) (dual of [157, 8, 113]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁴⁸, 153, F₄, 112) (dual of [153, 5, 113]-code), but
      - residual code [i] would yield linear OA(4³⁶, 40, F₄, 28) (dual of [40, 4, 29]-code), but
        residual code [i] would yield linear OA(4⁸, 11, F₄, 7) (dual of [11, 3, 8]-code), but
        improvement by Maruta on the Griesmer bound [i]
    2. OA(4⁸, 157, S₄, 4), but
      - discarding factors would yield OA(4⁸, 121, S₄, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 65704 > 4⁸ [i]

(157−120, 157, 244)-Net in Base 4 — Upper bound on s

There is no (37, 157, 245)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38850 600616 758101 183948 065073 257601 788274 545590 474029 874018 754823 738748 900117 435591 782161 584760 > 4¹⁵⁷ [i]