Best Known (45−21, 45, s)-Nets in Base 49
(45−21, 45, 344)-Net over F49 — Constructive and digital
Digital (24, 45, 344)-net over F49, using
- t-expansion [i] based on digital (21, 45, 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
(45−21, 45, 1347)-Net over F49 — Digital
Digital (24, 45, 1347)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4945, 1347, F49, 21) (dual of [1347, 1302, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4945, 2415, F49, 21) (dual of [2415, 2370, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(4941, 2401, F49, 21) (dual of [2401, 2360, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(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(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(4945, 2415, F49, 21) (dual of [2415, 2370, 22]-code), using
(45−21, 45, 2579865)-Net in Base 49 — Upper bound on s
There is no (24, 45, 2579866)-net in base 49, because
- 1 times m-reduction [i] would yield (24, 44, 2579866)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 233 683881 053935 675451 525713 306442 165061 439241 279091 963581 049498 124672 273345 > 4944 [i]