Best Known (37−24, 37, s)-Nets in Base 3
(37−24, 37, 24)-Net over F3 — Constructive and digital
Digital (13, 37, 24)-net over F3, using
- net from sequence [i] based on digital (13, 23)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 13 and N(F) ≥ 24, using
(37−24, 37, 61)-Net over F3 — Upper bound on s (digital)
There is no digital (13, 37, 62)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(337, 62, F3, 24) (dual of [62, 25, 25]-code), but
- construction Y1 [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
- OA(325, 62, S3, 16), but
- the Rao or (dual) Hamming bound shows that M ≥ 932661 812025 > 325 [i]
- linear OA(336, 46, F3, 24) (dual of [46, 10, 25]-code), but
- construction Y1 [i] would yield
(37−24, 37, 64)-Net in Base 3 — Upper bound on s
There is no (13, 37, 65)-net in base 3, because
- extracting embedded orthogonal array [i] would yield OA(337, 65, S3, 24), but
- the linear programming bound shows that M ≥ 426936 452476 178679 850933 929308 646861 912941 373333 / 862133 318649 022280 698526 213525 > 337 [i]