Best Known (6, 14, s)-Nets in Base 49
(6, 14, 150)-Net over F49 — Constructive and digital
Digital (6, 14, 150)-net over F49, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 50)-net over F49, using
- digital (0, 4, 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, 8, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
(6, 14, 244)-Net over F49 — Digital
Digital (6, 14, 244)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4914, 244, F49, 8) (dual of [244, 230, 9]-code), using
- construction XX applied to C1 = C([239,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([239,6]) [i] based on
- linear OA(4912, 240, F49, 7) (dual of [240, 228, 8]-code), using the BCH-code C(I) with length 240 | 492−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [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(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(4910, 240, F49, 6) (dual of [240, 230, 7]-code), using the expurgated narrow-sense BCH-code C(I) with length 240 | 492−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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([239,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([239,6]) [i] based on
(6, 14, 37973)-Net in Base 49 — Upper bound on s
There is no (6, 14, 37974)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 460014 500419 591736 942209 > 4914 [i]