Best Known (30, 30+24, s)-Nets in Base 49
(30, 30+24, 344)-Net over F49 — Constructive and digital
Digital (30, 54, 344)-net over F49, using
- t-expansion [i] based on digital (21, 54, 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
(30, 30+24, 2215)-Net over F49 — Digital
Digital (30, 54, 2215)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4954, 2215, F49, 24) (dual of [2215, 2161, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4954, 2424, F49, 24) (dual of [2424, 2370, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- linear OA(4947, 2401, F49, 24) (dual of [2401, 2354, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(497, 23, F49, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,49)), using
- discarding factors / shortening the dual code based on linear OA(497, 49, F49, 7) (dual of [49, 42, 8]-code or 49-arc in PG(6,49)), using
- Reed–Solomon code RS(42,49) [i]
- discarding factors / shortening the dual code based on linear OA(497, 49, F49, 7) (dual of [49, 42, 8]-code or 49-arc in PG(6,49)), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(4954, 2424, F49, 24) (dual of [2424, 2370, 25]-code), using
(30, 30+24, 4446332)-Net in Base 49 — Upper bound on s
There is no (30, 54, 4446333)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 18 646118 815090 122308 227409 246835 574395 410139 144903 433489 884322 307873 018042 393663 838947 659585 > 4954 [i]