MinT - Best Known (133−96, 133, s)-Nets in Base 4

Best Known (133−96, 133, s)-Nets in Base 4

(133−96, 133, 56)-Net over F₄ — Constructive and digital

Digital (37, 133, 56)-net over F₄, using

t-expansion [i] based on digital (33, 133, 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]

(133−96, 133, 66)-Net over F₄ — Digital

Digital (37, 133, 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]

(133−96, 133, 177)-Net over F₄ — Upper bound on s (digital)

There is no digital (37, 133, 178)-net over F₄, because

1 times m-reduction [i] would yield digital (37, 132, 178)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹³², 178, F₄, 95) (dual of [178, 46, 96]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(4¹³¹, 149, S₄, 95), but
      - the linear programming bound shows that MÂ â‰¥ 33368 039747 905148 197531 005626 408113 677562 887057 731053 159021 618938 897974 262171 069449 240576 / 3975 943153 > 4¹³¹ [i]
    2. OA(4⁴⁶, 178, S₄, 29), but
      - discarding factors would yield OA(4⁴⁶, 174, S₄, 29), but
        the linear programming bound shows that MÂ â‰¥ 8506 289708 537907 430011 501124 929257 400094 961052 942336 000000 / 1 703346 550862 338223 360728 253441 > 4⁴⁶ [i]

(133−96, 133, 253)-Net in Base 4 — Upper bound on s

There is no (37, 133, 254)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 130 815563 503533 086885 665234 542195 222323 121050 560166 024797 729391 702379 761131 524193 > 4¹³³ [i]