Best Known (8, 41, s)-Nets in Base 5
(8, 41, 23)-Net over F5 — Constructive and digital
Digital (8, 41, 23)-net over F5, using
- net from sequence [i] based on digital (8, 22)-sequence over F5, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F5 with g(F) = 7, N(F) = 22, and 1 place with degree 2 [i] based on function field F/F5 with g(F) = 7 and N(F) ≥ 22, using an explicitly constructive algebraic function field [i]
(8, 41, 51)-Net over F5 — Upper bound on s (digital)
There is no digital (8, 41, 52)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(541, 52, F5, 33) (dual of [52, 11, 34]-code), but
- construction Y1 [i] would yield
- linear OA(540, 44, F5, 33) (dual of [44, 4, 34]-code), but
- OA(511, 52, S5, 8), but
- discarding factors would yield OA(511, 48, S5, 8), but
- the Rao or (dual) Hamming bound shows that M ≥ 50 937665 > 511 [i]
- discarding factors would yield OA(511, 48, S5, 8), but
- construction Y1 [i] would yield
(8, 41, 53)-Net in Base 5 — Upper bound on s
There is no (8, 41, 54)-net in base 5, because
- extracting embedded orthogonal array [i] would yield OA(541, 54, S5, 33), but
- the linear programming bound shows that M ≥ 12 820265 737900 626845 657825 469970 703125 / 255 238153 > 541 [i]