MinT - Best Known (9, 36, s)-Nets in Base 4

Best Known (9, 36, s)-Nets in Base 4

(9, 36, 22)-Net over F₄ — Constructive and digital

Digital (9, 36, 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]

(9, 36, 26)-Net over F₄ — Digital

Digital (9, 36, 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]

(9, 36, 50)-Net over F₄ — Upper bound on s (digital)

There is no digital (9, 36, 51)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4³⁶, 51, F₄, 27) (dual of [51, 15, 28]-code), but
- â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(9, 36, 54)-Net in Base 4 — Upper bound on s

There is no (9, 36, 55)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4³⁶, 55, S₄, 27), but
- the linear programming bound shows that MÂ â‰¥ 930883 572540 981681 400519 327744 / 173 327869 > 4³⁶ [i]