Best Known (50−40, 50, s)-Nets in Base 5
(50−40, 50, 26)-Net over F5 — Constructive and digital
Digital (10, 50, 26)-net over F5, using
- t-expansion [i] based on digital (9, 50, 26)-net over F5, using
- net from sequence [i] based on digital (9, 25)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 9 and N(F) ≥ 26, using
- net from sequence [i] based on digital (9, 25)-sequence over F5, using
(50−40, 50, 27)-Net over F5 — Digital
Digital (10, 50, 27)-net over F5, using
- net from sequence [i] based on digital (10, 26)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 10 and N(F) ≥ 27, using
(50−40, 50, 58)-Net over F5 — Upper bound on s (digital)
There is no digital (10, 50, 59)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(550, 59, F5, 40) (dual of [59, 9, 41]-code), but
- construction Y1 [i] would yield
- linear OA(549, 53, F5, 40) (dual of [53, 4, 41]-code), but
- residual code [i] would yield linear OA(59, 12, F5, 8) (dual of [12, 3, 9]-code), but
- OA(59, 59, S5, 6), but
- discarding factors would yield OA(59, 58, S5, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 2 001465 > 59 [i]
- discarding factors would yield OA(59, 58, S5, 6), but
- linear OA(549, 53, F5, 40) (dual of [53, 4, 41]-code), but
- construction Y1 [i] would yield
(50−40, 50, 62)-Net in Base 5 — Upper bound on s
There is no (10, 50, 63)-net in base 5, because
- extracting embedded orthogonal array [i] would yield OA(550, 63, S5, 40), but
- the linear programming bound shows that M ≥ 176003 656093 826066 353358 328342 437744 140625 / 1 932781 > 550 [i]