Best Known (55−27, 55, s)-Nets in Base 49
(55−27, 55, 344)-Net over F49 — Constructive and digital
Digital (28, 55, 344)-net over F49, using
- t-expansion [i] based on digital (21, 55, 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
(55−27, 55, 1092)-Net over F49 — Digital
Digital (28, 55, 1092)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4955, 1092, F49, 2, 27) (dual of [(1092, 2), 2129, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4955, 1204, F49, 2, 27) (dual of [(1204, 2), 2353, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4955, 2408, F49, 27) (dual of [2408, 2353, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4955, 2409, F49, 27) (dual of [2409, 2354, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- linear OA(4953, 2401, F49, 27) (dual of [2401, 2348, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 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(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(26) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(4955, 2409, F49, 27) (dual of [2409, 2354, 28]-code), using
- OOA 2-folding [i] based on linear OA(4955, 2408, F49, 27) (dual of [2408, 2353, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(4955, 1204, F49, 2, 27) (dual of [(1204, 2), 2353, 28]-NRT-code), using
(55−27, 55, 1238731)-Net in Base 49 — Upper bound on s
There is no (28, 55, 1238732)-net in base 49, because
- 1 times m-reduction [i] would yield (28, 54, 1238732)-net in base 49, but
- 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]