MinT - Best Known (222−161, 222, s)-Nets in Base 3

Best Known (222−161, 222, s)-Nets in Base 3

(222−161, 222, 48)-Net over F₃ — Constructive and digital

Digital (61, 222, 48)-net over F₃, using

t-expansion [i] based on digital (45, 222, 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]

(222−161, 222, 64)-Net over F₃ — Digital

Digital (61, 222, 64)-net over F₃, using

t-expansion [i] based on digital (49, 222, 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]

(222−161, 222, 192)-Net over F₃ — Upper bound on s (digital)

There is no digital (61, 222, 193)-net over F₃, because

35 times m-reduction [i] would yield digital (61, 187, 193)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁷, 193, F₃, 126) (dual of [193, 6, 127]-code), but
  - residual code [i] would yield linear OA(3⁶¹, 66, F₃, 42) (dual of [66, 5, 43]-code), but
    - residual code [i] would yield linear OA(3¹⁹, 23, F₃, 14) (dual of [23, 4, 15]-code), but
      - 2 times truncation [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]

(222−161, 222, 196)-Net in Base 3 — Upper bound on s

There is no (61, 222, 197)-net in base 3, because

30 times m-reduction [i] would yield (61, 192, 197)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁹², 197, S₃, 131), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 1093 061682 616768 598101 980749 118434 678309 602685 816438 255039 403134 728775 682721 408160 470718 926107 / 22 > 3¹⁹² [i]