Best Known (22, 22+22, s)-Nets in Base 49
(22, 22+22, 344)-Net over F49 — Constructive and digital
Digital (22, 44, 344)-net over F49, using
- t-expansion [i] based on digital (21, 44, 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
(22, 22+22, 891)-Net over F49 — Digital
Digital (22, 44, 891)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4944, 891, F49, 2, 22) (dual of [(891, 2), 1738, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4944, 1203, F49, 2, 22) (dual of [(1203, 2), 2362, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4944, 2406, F49, 22) (dual of [2406, 2362, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [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(4939, 2401, F49, 20) (dual of [2401, 2362, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(491, 5, F49, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(4944, 2406, F49, 22) (dual of [2406, 2362, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(4944, 1203, F49, 2, 22) (dual of [(1203, 2), 2362, 23]-NRT-code), using
(22, 22+22, 589594)-Net in Base 49 — Upper bound on s
There is no (22, 44, 589595)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 233 685611 704116 238844 583803 404681 740531 136759 341739 844870 942335 805271 125425 > 4944 [i]