Best Known (49−23, 49, s)-Nets in Base 49
(49−23, 49, 344)-Net over F49 — Constructive and digital
Digital (26, 49, 344)-net over F49, using
- t-expansion [i] based on digital (21, 49, 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
(49−23, 49, 1310)-Net over F49 — Digital
Digital (26, 49, 1310)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4949, 1310, F49, 23) (dual of [1310, 1261, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4949, 2415, F49, 23) (dual of [2415, 2366, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [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(4935, 2401, F49, 18) (dual of [2401, 2366, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4949, 2415, F49, 23) (dual of [2415, 2366, 24]-code), using
(49−23, 49, 2427577)-Net in Base 49 — Upper bound on s
There is no (26, 49, 2427578)-net in base 49, because
- 1 times m-reduction [i] would yield (26, 48, 2427578)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 1347 141784 422308 431726 753081 529229 638857 988172 862087 942388 961535 827142 303424 282785 > 4948 [i]