MinT - Best Known (137−100, 137, s)-Nets in Base 4

Best Known (137−100, 137, s)-Nets in Base 4

(137−100, 137, 56)-Net over F₄ — Constructive and digital

Digital (37, 137, 56)-net over F₄, using

t-expansion [i] based on digital (33, 137, 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]

(137−100, 137, 66)-Net over F₄ — Digital

Digital (37, 137, 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]

(137−100, 137, 166)-Net over F₄ — Upper bound on s (digital)

There is no digital (37, 137, 167)-net over F₄, because

1 times m-reduction [i] would yield digital (37, 136, 167)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹³⁶, 167, F₄, 99) (dual of [167, 31, 100]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(4¹³⁵, 149, S₄, 99), but
      - the linear programming bound shows that MÂ â‰¥ 2 040610 487096 181427 758937 498317 933435 839690 422249 792187 355243 784188 300883 648114 289820 041216 / 970 822125 > 4¹³⁵ [i]
    2. OA(4³¹, 167, S₄, 18), but
      - the linear programming bound shows that MÂ â‰¥ 10 709160 698850 412098 268143 898591 232000 / 2 251884 149854 892341 > 4³¹ [i]

(137−100, 137, 250)-Net in Base 4 — Upper bound on s

There is no (37, 137, 251)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 31558 100284 923326 612791 076812 691713 298820 807468 918905 577156 351217 331768 901288 339721 > 4¹³⁷ [i]