Best Known (53−25, 53, s)-Nets in Base 49
(53−25, 53, 344)-Net over F49 — Constructive and digital
Digital (28, 53, 344)-net over F49, using
- t-expansion [i] based on digital (21, 53, 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
(53−25, 53, 1291)-Net over F49 — Digital
Digital (28, 53, 1291)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4953, 1291, F49, 25) (dual of [1291, 1238, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4953, 2415, F49, 25) (dual of [2415, 2362, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(4949, 2401, F49, 25) (dual of [2401, 2352, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(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(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(4953, 2415, F49, 25) (dual of [2415, 2362, 26]-code), using
(53−25, 53, 2324353)-Net in Base 49 — Upper bound on s
There is no (28, 53, 2324354)-net in base 49, because
- 1 times m-reduction [i] would yield (28, 52, 2324354)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 7766 011697 645431 572832 660577 921493 958277 475807 155251 978595 865656 108886 697852 695151 991425 > 4952 [i]