MinT - Best Known (226−180, 226, s)-Nets in Base 4

Best Known (226−180, 226, s)-Nets in Base 4

(226−180, 226, 56)-Net over F₄ — Constructive and digital

Digital (46, 226, 56)-net over F₄, using

t-expansion [i] based on digital (33, 226, 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]

(226−180, 226, 81)-Net over F₄ — Digital

Digital (46, 226, 81)-net over F₄, using

net from sequence [i] based on digital (46, 80)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
  - Drinfeld modules of rank 1 [i]

(226−180, 226, 193)-Net over F₄ — Upper bound on s (digital)

There is no digital (46, 226, 194)-net over F₄, because

36 times m-reduction [i] would yield digital (46, 190, 194)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹⁰, 194, F₄, 144) (dual of [194, 4, 145]-code), but
  - Griesmer bound [i]

(226−180, 226, 198)-Net in Base 4 — Upper bound on s

There is no (46, 226, 199)-net in base 4, because

31 times m-reduction [i] would yield (46, 195, 199)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁹⁵, 199, S₄, 149), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 80695 308690 215893 426747 474125 094121 072803 306025 913234 775958 104891 895238 188026 287332 176417 290004 307232 371974 124148 359168 / 25 > 4¹⁹⁵ [i]