MinT - Best Known (54, 54+186, s)-Nets in Base 3

Best Known (54, 54+186, s)-Nets in Base 3

(54, 54+186, 48)-Net over F₃ — Constructive and digital

Digital (54, 240, 48)-net over F₃, using

t-expansion [i] based on digital (45, 240, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(54, 54+186, 64)-Net over F₃ — Digital

Digital (54, 240, 64)-net over F₃, using

t-expansion [i] based on digital (49, 240, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(54, 54+186, 170)-Net over F₃ — Upper bound on s (digital)

There is no digital (54, 240, 171)-net over F₃, because

78 times m-reduction [i] would yield digital (54, 162, 171)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶², 171, F₃, 108) (dual of [171, 9, 109]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁶¹, 167, F₃, 108) (dual of [167, 6, 109]-code), but
      - residual code [i] would yield linear OA(3⁵³, 58, F₃, 36) (dual of [58, 5, 37]-code), but
        residual code [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(3⁹, 171, S₃, 4), but
      - discarding factors would yield OA(3⁹, 100, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 20001 > 3⁹ [i]

(54, 54+186, 175)-Net in Base 3 — Upper bound on s

There is no (54, 240, 176)-net in base 3, because

69 times m-reduction [i] would yield (54, 171, 176)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁷¹, 176, S₃, 117), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 313487 028995 334947 898311 636710 352105 968826 343219 733319 851858 295246 737190 712070 115307 / 59 > 3¹⁷¹ [i]