Best Known (41−19, 41, s)-Nets in Base 49
(41−19, 41, 344)-Net over F49 — Constructive and digital
Digital (22, 41, 344)-net over F49, using
- t-expansion [i] based on digital (21, 41, 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
(41−19, 41, 1410)-Net over F49 — Digital
Digital (22, 41, 1410)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4941, 1410, F49, 19) (dual of [1410, 1369, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4941, 2415, F49, 19) (dual of [2415, 2374, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(4937, 2401, F49, 19) (dual of [2401, 2364, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(494, 14, F49, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,49)), using
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- Reed–Solomon code RS(45,49) [i]
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4941, 2415, F49, 19) (dual of [2415, 2374, 20]-code), using
(41−19, 41, 2808614)-Net in Base 49 — Upper bound on s
There is no (22, 41, 2808615)-net in base 49, because
- 1 times m-reduction [i] would yield (22, 40, 2808615)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 40 536274 267493 203130 821661 096227 015950 008278 938467 023397 508856 545745 > 4940 [i]