MinT - Best Known (56, 56+116, s)-Nets in Base 3

Best Known (56, 56+116, s)-Nets in Base 3

(56, 56+116, 48)-Net over F₃ — Constructive and digital

Digital (56, 172, 48)-net over F₃, using

t-expansion [i] based on digital (45, 172, 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]

(56, 56+116, 64)-Net over F₃ — Digital

Digital (56, 172, 64)-net over F₃, using

t-expansion [i] based on digital (49, 172, 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]

(56, 56+116, 178)-Net over F₃ — Upper bound on s (digital)

There is no digital (56, 172, 179)-net over F₃, because

2 times m-reduction [i] would yield digital (56, 170, 179)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁰, 179, F₃, 114) (dual of [179, 9, 115]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁶⁹, 175, F₃, 114) (dual of [175, 6, 115]-code), but
      - residual code [i] would yield linear OA(3⁵⁵, 60, F₃, 38) (dual of [60, 5, 39]-code), but
        2 times truncation [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, 179, 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]

(56, 56+116, 238)-Net in Base 3 — Upper bound on s

There is no (56, 172, 239)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12616 335239 283312 211913 384123 389096 180107 254266 687968 412532 573871 733311 382759 263085 > 3¹⁷² [i]