MinT - Best Known (135−97, 135, s)-Nets in Base 4

Best Known (135−97, 135, s)-Nets in Base 4

(135−97, 135, 56)-Net over F₄ — Constructive and digital

Digital (38, 135, 56)-net over F₄, using

t-expansion [i] based on digital (33, 135, 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]

(135−97, 135, 66)-Net over F₄ — Digital

Digital (38, 135, 66)-net over F₄, using

t-expansion [i] based on digital (37, 135, 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]

(135−97, 135, 243)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 135, 244)-net over F₄, because

1 times m-reduction [i] would yield digital (38, 134, 244)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹³⁴, 244, F₄, 96) (dual of [244, 110, 97]-code), but
  - residual code [i] would yield OA(4³⁸, 147, S₄, 24), but
    - the linear programming bound shows that MÂ â‰¥ 5459 111453 470077 053881 598601 691985 346560 000000 / 69230 222284 404391 824599 > 4³⁸ [i]

(135−97, 135, 261)-Net in Base 4 — Upper bound on s

There is no (38, 135, 262)-net in base 4, because

1 times m-reduction [i] would yield (38, 134, 262)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 483 342403 684785 699524 672543 899174 961177 596476 333563 747667 246501 750675 852595 428112 > 4¹³⁴ [i]