MinT - Best Known (37−28, 37, s)-Nets in Base 4

Best Known (37−28, 37, s)-Nets in Base 4

(37−28, 37, 22)-Net over F₄ — Constructive and digital

Digital (9, 37, 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]

(37−28, 37, 26)-Net over F₄ — Digital

Digital (9, 37, 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]

(37−28, 37, 45)-Net over F₄ — Upper bound on s (digital)

There is no digital (9, 37, 46)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4³⁷, 46, F₄, 28) (dual of [46, 9, 29]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(4³⁶, 40, F₄, 28) (dual of [40, 4, 29]-code), but
    - residual code [i] would yield linear OA(4⁸, 11, F₄, 7) (dual of [11, 3, 8]-code), but
      - improvement by Maruta on the Griesmer bound [i]
  2. OA(4⁹, 46, S₄, 6), but
    - discarding factors would yield OA(4⁹, 40, S₄, 6), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 273901 > 4⁹ [i]

(37−28, 37, 50)-Net in Base 4 — Upper bound on s

There is no (9, 37, 51)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4³⁷, 51, S₄, 28), but
- the linear programming bound shows that MÂ â‰¥ 12 654532 566584 978515 087572 598784 / 545 483939 > 4³⁷ [i]