Best Known (45−22, 45, s)-Nets in Base 49
(45−22, 45, 344)-Net over F49 — Constructive and digital
Digital (23, 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
(45−22, 45, 1096)-Net over F49 — Digital
Digital (23, 45, 1096)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4945, 1096, F49, 2, 22) (dual of [(1096, 2), 2147, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4945, 1204, F49, 2, 22) (dual of [(1204, 2), 2363, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4945, 2408, F49, 22) (dual of [2408, 2363, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4945, 2409, F49, 22) (dual of [2409, 2364, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- 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(4937, 2401, F49, 19) (dual of [2401, 2364, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(4945, 2409, F49, 22) (dual of [2409, 2364, 23]-code), using
- OOA 2-folding [i] based on linear OA(4945, 2408, F49, 22) (dual of [2408, 2363, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(4945, 1204, F49, 2, 22) (dual of [(1204, 2), 2363, 23]-NRT-code), using
(45−22, 45, 839863)-Net in Base 49 — Upper bound on s
There is no (23, 45, 839864)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 11450 547780 719704 443415 774401 260215 954662 778288 386809 474580 699762 069850 013569 > 4945 [i]