Best Known (7, 7+9, s)-Nets in Base 49
(7, 7+9, 150)-Net over F49 — Constructive and digital
Digital (7, 16, 150)-net over F49, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 3, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- digital (0, 4, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
- digital (0, 9, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
- digital (0, 3, 50)-net over F49, using
(7, 7+9, 244)-Net over F49 — Digital
Digital (7, 16, 244)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4916, 244, F49, 9) (dual of [244, 228, 10]-code), using
- construction XX applied to C1 = C([239,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([239,7]) [i] based on
- linear OA(4914, 240, F49, 8) (dual of [240, 226, 9]-code), using the BCH-code C(I) with length 240 | 492−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(4914, 240, F49, 8) (dual of [240, 226, 9]-code), using the expurgated narrow-sense BCH-code C(I) with length 240 | 492−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(4916, 240, F49, 9) (dual of [240, 224, 10]-code), using the BCH-code C(I) with length 240 | 492−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4912, 240, F49, 7) (dual of [240, 228, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 240 | 492−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([239,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([239,7]) [i] based on
(7, 7+9, 100470)-Net in Base 49 — Upper bound on s
There is no (7, 16, 100471)-net in base 49, because
- 1 times m-reduction [i] would yield (7, 15, 100471)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 22 539484 718014 464027 750721 > 4915 [i]