MinT - Best Known (103−66, 103, s)-Nets in Base 3

Best Known (103−66, 103, s)-Nets in Base 3

(103−66, 103, 38)-Net over F₃ — Constructive and digital

Digital (37, 103, 38)-net over F₃, using

t-expansion [i] based on digital (32, 103, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(103−66, 103, 52)-Net over F₃ — Digital

Digital (37, 103, 52)-net over F₃, using

net from sequence [i] based on digital (37, 51)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 37 and N(F) ≥ 52, using
  - a curve from the manYPoints database [i]

(103−66, 103, 123)-Net over F₃ — Upper bound on s (digital)

There is no digital (37, 103, 124)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁰³, 124, F₃, 66) (dual of [124, 21, 67]-code), but
- â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(103−66, 103, 124)-Net in Base 3 — Upper bound on s

There is no (37, 103, 125)-net in base 3, because

1 times m-reduction [i] would yield (37, 102, 125)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁰², 125, S₃, 65), but
  - the linear programming bound shows that MÂ â‰¥ 82769 748454 520344 172235 733433 653475 678772 427150 151046 492171 / 16360 421572 > 3¹⁰² [i]