MinT - Best Known (231−162, 231, s)-Nets in Base 3

Best Known (231−162, 231, s)-Nets in Base 3

(231−162, 231, 48)-Net over F₃ — Constructive and digital

Digital (69, 231, 48)-net over F₃, using

t-expansion [i] based on digital (45, 231, 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]

(231−162, 231, 82)-Net over F₃ — Digital

Digital (69, 231, 82)-net over F₃, using

net from sequence [i] based on digital (69, 81)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
  - Drinfeld modules of rank 1 [i]

(231−162, 231, 217)-Net over F₃ — Upper bound on s (digital)

There is no digital (69, 231, 218)-net over F₃, because

21 times m-reduction [i] would yield digital (69, 210, 218)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²¹⁰, 218, F₃, 141) (dual of [218, 8, 142]-code), but
  - residual code [i] would yield linear OA(3⁶⁹, 76, F₃, 47) (dual of [76, 7, 48]-code), but
    - 2 times truncation [i] would yield linear OA(3⁶⁷, 74, F₃, 45) (dual of [74, 7, 46]-code), but
      - residual code [i] would yield linear OA(3²², 28, F₃, 15) (dual of [28, 6, 16]-code), but
        â€œHHMâ€ bound on codes from Brouwerâ€™s database [i]

(231−162, 231, 220)-Net in Base 3 — Upper bound on s

There is no (69, 231, 221)-net in base 3, because

15 times m-reduction [i] would yield (69, 216, 221)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²¹⁶, 221, S₃, 147), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 926 138713 099787 670959 935798 024513 966701 772293 499227 988263 405269 197039 529170 894882 252068 039219 702299 428401 / 74 > 3²¹⁶ [i]