MinT - Best Known (62, 62+132, s)-Nets in Base 3

Best Known (62, 62+132, s)-Nets in Base 3

(62, 62+132, 48)-Net over F₃ — Constructive and digital

Digital (62, 194, 48)-net over F₃, using

t-expansion [i] based on digital (45, 194, 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]

(62, 62+132, 64)-Net over F₃ — Digital

Digital (62, 194, 64)-net over F₃, using

t-expansion [i] based on digital (49, 194, 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]

(62, 62+132, 195)-Net over F₃ — Upper bound on s (digital)

There is no digital (62, 194, 196)-net over F₃, because

6 times m-reduction [i] would yield digital (62, 188, 196)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁸, 196, F₃, 126) (dual of [196, 8, 127]-code), but
  - residual code [i] would yield linear OA(3⁶², 69, F₃, 42) (dual of [69, 7, 43]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(62, 62+132, 260)-Net in Base 3 — Upper bound on s

There is no (62, 194, 261)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 402 567957 807042 375808 442145 748766 221815 158815 623721 863151 483442 157645 217500 144826 742845 115217 > 3¹⁹⁴ [i]