MinT - Best Known (42−27, 42, s)-Nets in Base 3

Best Known (42−27, 42, s)-Nets in Base 3

(42−27, 42, 28)-Net over F₃ — Constructive and digital

Digital (15, 42, 28)-net over F₃, using

net from sequence [i] based on digital (15, 27)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
  - an explicitly constructive algebraic function field [i]

(42−27, 42, 60)-Net over F₃ — Upper bound on s (digital)

There is no digital (15, 42, 61)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁴², 61, F₃, 27) (dual of [61, 19, 28]-code), but
- residual code [i] would yield linear OA(3¹⁵, 33, F₃, 9) (dual of [33, 18, 10]-code), but
  - â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(42−27, 42, 71)-Net in Base 3 — Upper bound on s

There is no (15, 42, 72)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁴², 72, S₃, 27), but
- the linear programming bound shows that MÂ â‰¥ 459 570515 671288 975520 197273 080900 198455 977122 916419 085479 / 4 115300 885402 156269 619771 733358 710080 > 3⁴² [i]