Best Known (45−20, 45, s)-Nets in Base 49
(45−20, 45, 344)-Net over F49 — Constructive and digital
Digital (25, 45, 344)-net over F49, using
- t-expansion [i] based on digital (21, 45, 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
(45−20, 45, 2122)-Net over F49 — Digital
Digital (25, 45, 2122)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4945, 2122, F49, 20) (dual of [2122, 2077, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4945, 2421, F49, 20) (dual of [2421, 2376, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(12) [i] based on
- 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(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(496, 20, F49, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,49)), using
- discarding factors / shortening the dual code based on linear OA(496, 49, F49, 6) (dual of [49, 43, 7]-code or 49-arc in PG(5,49)), using
- Reed–Solomon code RS(43,49) [i]
- discarding factors / shortening the dual code based on linear OA(496, 49, F49, 6) (dual of [49, 43, 7]-code or 49-arc in PG(5,49)), using
- construction X applied to Ce(19) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4945, 2421, F49, 20) (dual of [2421, 2376, 21]-code), using
(45−20, 45, 3807298)-Net in Base 49 — Upper bound on s
There is no (25, 45, 3807299)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 11450 502851 140941 949455 724544 733721 503165 642043 658214 373235 033692 746665 450401 > 4945 [i]