Best Known (51−23, 51, s)-Nets in Base 49
(51−23, 51, 344)-Net over F49 — Constructive and digital
Digital (28, 51, 344)-net over F49, using
- t-expansion [i] based on digital (21, 51, 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
(51−23, 51, 1902)-Net over F49 — Digital
Digital (28, 51, 1902)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4951, 1902, F49, 23) (dual of [1902, 1851, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4951, 2421, F49, 23) (dual of [2421, 2370, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(15) [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(4931, 2401, F49, 16) (dual of [2401, 2370, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(22) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(4951, 2421, F49, 23) (dual of [2421, 2370, 24]-code), using
(51−23, 51, 4925858)-Net in Base 49 — Upper bound on s
There is no (28, 51, 4925859)-net in base 49, because
- 1 times m-reduction [i] would yield (28, 50, 4925859)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 3 234481 735729 542302 506117 807319 928659 238086 950555 181236 319136 994304 687351 044226 689585 > 4950 [i]