MinT - Best Known (34−25, 34, s)-Nets in Base 4

Best Known (34−25, 34, s)-Nets in Base 4

(34−25, 34, 22)-Net over F₄ — Constructive and digital

Digital (9, 34, 22)-net over F₄, using

net from sequence [i] based on digital (9, 21)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
  - an explicitly constructive algebraic function field [i]

(34−25, 34, 26)-Net over F₄ — Digital

Digital (9, 34, 26)-net over F₄, using

net from sequence [i] based on digital (9, 25)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 26, using
  - Drinfeld modules of rank 1 [i]

(34−25, 34, 63)-Net over F₄ — Upper bound on s (digital)

There is no digital (9, 34, 64)-net over F₄, because

1 times m-reduction [i] would yield digital (9, 33, 64)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4³³, 64, F₄, 24) (dual of [64, 31, 25]-code), but
  - residual code [i] would yield OA(4⁹, 39, S₄, 6), but
    - the linear programming bound shows that MÂ â‰¥ 504 832000 / 1843 > 4⁹ [i]

(34−25, 34, 65)-Net in Base 4 — Upper bound on s

There is no (9, 34, 66)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4³⁴, 66, S₄, 25), but
- the linear programming bound shows that MÂ â‰¥ 604 960001 283991 112309 250407 192187 024483 029217 018440 762263 061173 021045 360659 791872 / 1 890179 221099 436971 006558 191042 915661 275690 944803 178900 711059 > 4³⁴ [i]