Best Known (54−26, 54, s)-Nets in Base 49
(54−26, 54, 344)-Net over F49 — Constructive and digital
Digital (28, 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
(54−26, 54, 1206)-Net over F49 — Digital
Digital (28, 54, 1206)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4954, 1206, F49, 2, 26) (dual of [(1206, 2), 2358, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4954, 2412, F49, 26) (dual of [2412, 2358, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(4951, 2401, F49, 26) (dual of [2401, 2350, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(493, 11, F49, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,49) or 11-cap in PG(2,49)), using
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- Reed–Solomon code RS(46,49) [i]
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(4954, 2412, F49, 26) (dual of [2412, 2358, 27]-code), using
(54−26, 54, 1238731)-Net in Base 49 — Upper bound on s
There is no (28, 54, 1238732)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 18 646185 000433 870249 242589 701695 578727 366313 725434 558361 682063 543750 050574 730993 801237 771585 > 4954 [i]