MinT - Best Known (61, 61+130, s)-Nets in Base 3

Best Known (61, 61+130, s)-Nets in Base 3

(61, 61+130, 48)-Net over F₃ — Constructive and digital

Digital (61, 191, 48)-net over F₃, using

t-expansion [i] based on digital (45, 191, 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]

(61, 61+130, 64)-Net over F₃ — Digital

Digital (61, 191, 64)-net over F₃, using

t-expansion [i] based on digital (49, 191, 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]

(61, 61+130, 192)-Net over F₃ — Upper bound on s (digital)

There is no digital (61, 191, 193)-net over F₃, because

4 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]

(61, 61+130, 256)-Net in Base 3 — Upper bound on s

There is no (61, 191, 257)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15 152385 903352 434556 870797 844621 355991 808869 130425 017649 756513 435596 535817 511620 885231 155459 > 3¹⁹¹ [i]