MinT - Best Known (8, 8+27, s)-Nets in Base 4

Best Known (8, 8+27, s)-Nets in Base 4

(8, 8+27, 21)-Net over F₄ — Constructive and digital

Digital (8, 35, 21)-net over F₄, using

t-expansion [i] based on digital (7, 35, 21)-net over F₄, using
- net from sequence [i] based on digital (7, 20)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(8, 8+27, 40)-Net over F₄ — Upper bound on s (digital)

There is no digital (8, 35, 41)-net over F₄, because

3 times m-reduction [i] would yield digital (8, 32, 41)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4³², 41, F₄, 24) (dual of [41, 9, 25]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4³¹, 35, F₄, 24) (dual of [35, 4, 25]-code), but
      - residual code [i] would yield linear OA(4⁷, 10, F₄, 6) (dual of [10, 3, 7]-code), but
        bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
    2. OA(4⁹, 41, 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]

(8, 8+27, 43)-Net in Base 4 — Upper bound on s

There is no (8, 35, 44)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4³⁵, 44, S₄, 27), but
- the linear programming bound shows that MÂ â‰¥ 136 608617 616453 096741 797888 / 83657 > 4³⁵ [i]