Best Known (24, 24+20, s)-Nets in Base 49
(24, 24+20, 344)-Net over F49 — Constructive and digital
Digital (24, 44, 344)-net over F49, using
- t-expansion [i] based on digital (21, 44, 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
(24, 24+20, 1708)-Net over F49 — Digital
Digital (24, 44, 1708)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4944, 1708, F49, 20) (dual of [1708, 1664, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4944, 2418, F49, 20) (dual of [2418, 2374, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [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(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(495, 17, F49, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,49)), using
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- Reed–Solomon code RS(44,49) [i]
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4944, 2418, F49, 20) (dual of [2418, 2374, 21]-code), using
(24, 24+20, 2579865)-Net in Base 49 — Upper bound on s
There is no (24, 44, 2579866)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 233 683881 053935 675451 525713 306442 165061 439241 279091 963581 049498 124672 273345 > 4944 [i]