MinT - Best Known (8, 8+24, s)-Nets in Base 4

Best Known (8, 8+24, s)-Nets in Base 4

(8, 8+24, 21)-Net over F₄ — Constructive and digital

Digital (8, 32, 21)-net over F₄, using

t-expansion [i] based on digital (7, 32, 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+24, 40)-Net over F₄ — Upper bound on s (digital)

There is no digital (8, 32, 41)-net over F₄, because

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+24, 52)-Net in Base 4 — Upper bound on s

There is no (8, 32, 53)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4³², 53, S₄, 24), but
- the linear programming bound shows that MÂ â‰¥ 1567 806561 917286 067162 737175 403400 527872 / 82 042521 136315 980625 > 4³² [i]