Best Known (12, 12+24, s)-Nets in Base 3
(12, 12+24, 20)-Net over F3 — Constructive and digital
Digital (12, 36, 20)-net over F3, using
- t-expansion [i] based on digital (11, 36, 20)-net over F3, using
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 9, N(F) = 19, and 1 place with degree 3 [i] based on function field F/F3 with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
(12, 12+24, 22)-Net over F3 — Digital
Digital (12, 36, 22)-net over F3, using
- net from sequence [i] based on digital (12, 21)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 12 and N(F) ≥ 22, using
(12, 12+24, 45)-Net over F3 — Upper bound on s (digital)
There is no digital (12, 36, 46)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(336, 46, F3, 24) (dual of [46, 10, 25]-code), but
- construction Y1 [i] would yield
- linear OA(335, 40, F3, 24) (dual of [40, 5, 25]-code), but
- “vE1†bound on codes from Brouwer’s database [i]
- OA(310, 46, S3, 6), but
- discarding factors would yield OA(310, 36, S3, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 59713 > 310 [i]
- discarding factors would yield OA(310, 36, S3, 6), but
- linear OA(335, 40, F3, 24) (dual of [40, 5, 25]-code), but
- construction Y1 [i] would yield
(12, 12+24, 53)-Net in Base 3 — Upper bound on s
There is no (12, 36, 54)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(336, 54, S3, 24), but
- the linear programming bound shows that M ≥ 646544 059356 453720 244620 792291 / 4 303819 609375 > 336 [i]