Best Known (51−25, 51, s)-Nets in Base 49
(51−25, 51, 344)-Net over F49 — Constructive and digital
Digital (26, 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
(51−25, 51, 1086)-Net over F49 — Digital
Digital (26, 51, 1086)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4951, 1086, F49, 2, 25) (dual of [(1086, 2), 2121, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4951, 1204, F49, 2, 25) (dual of [(1204, 2), 2357, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4951, 2408, F49, 25) (dual of [2408, 2357, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4951, 2409, F49, 25) (dual of [2409, 2358, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [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(4943, 2401, F49, 22) (dual of [2401, 2358, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(492, 8, F49, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,49)), using
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- Reed–Solomon code RS(47,49) [i]
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4951, 2409, F49, 25) (dual of [2409, 2358, 26]-code), using
- OOA 2-folding [i] based on linear OA(4951, 2408, F49, 25) (dual of [2408, 2357, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(4951, 1204, F49, 2, 25) (dual of [(1204, 2), 2357, 26]-NRT-code), using
(51−25, 51, 1215071)-Net in Base 49 — Upper bound on s
There is no (26, 51, 1215072)-net in base 49, because
- 1 times m-reduction [i] would yield (26, 50, 1215072)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 3 234496 639092 941677 203425 341686 296716 465000 233465 428777 065050 300445 138007 792656 103425 > 4950 [i]