Best Known (47−23, 47, s)-Nets in Base 49
(47−23, 47, 344)-Net over F49 — Constructive and digital
Digital (24, 47, 344)-net over F49, using
- t-expansion [i] based on digital (21, 47, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
(47−23, 47, 1090)-Net over F49 — Digital
Digital (24, 47, 1090)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4947, 1090, F49, 2, 23) (dual of [(1090, 2), 2133, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4947, 1204, F49, 2, 23) (dual of [(1204, 2), 2361, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4947, 2408, F49, 23) (dual of [2408, 2361, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4947, 2409, F49, 23) (dual of [2409, 2362, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(4945, 2401, F49, 23) (dual of [2401, 2356, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4939, 2401, F49, 20) (dual of [2401, 2362, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(492, 8, F49, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,49)), using
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- Reed–Solomon code RS(47,49) [i]
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(4947, 2409, F49, 23) (dual of [2409, 2362, 24]-code), using
- OOA 2-folding [i] based on linear OA(4947, 2408, F49, 23) (dual of [2408, 2361, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(4947, 1204, F49, 2, 23) (dual of [(1204, 2), 2361, 24]-NRT-code), using
(47−23, 47, 1196365)-Net in Base 49 — Upper bound on s
There is no (24, 47, 1196366)-net in base 49, because
- 1 times m-reduction [i] would yield (24, 46, 1196366)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 561078 280505 419202 998642 271414 905602 999467 066988 878858 085903 283125 360985 020385 > 4946 [i]