MinT - Best Known (37, 135, s)-Nets in Base 4

Best Known (37, 135, s)-Nets in Base 4

(37, 135, 56)-Net over F₄ — Constructive and digital

Digital (37, 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]

(37, 135, 66)-Net over F₄ — Digital

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]

(37, 135, 170)-Net over F₄ — Upper bound on s (digital)

There is no digital (37, 135, 171)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹³⁵, 171, F₄, 98) (dual of [171, 36, 99]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹³⁴, 149, S₄, 98), but
    - the linear programming bound shows that MÂ â‰¥ 16 522880 710265 097728 979611 544086 920533 682517 483034 531558 847934 994306 996193 933029 168694 427648 / 26940 869703 > 4¹³⁴ [i]
  2. OA(4³⁶, 171, S₄, 22), but
    - discarding factors would yield OA(4³⁶, 158, S₄, 22), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4882 374384 904669 137046 > 4³⁶ [i]

(37, 135, 252)-Net in Base 4 — Upper bound on s

There is no (37, 135, 253)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2240 390062 880074 181404 542300 298707 874496 700247 513298 053192 489677 380354 137105 381864 > 4¹³⁵ [i]