Best Known (16, 27, s)-Nets in Base 7
(16, 27, 108)-Net over F7 — Constructive and digital
Digital (16, 27, 108)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (11, 22, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 11, 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
- trace code for nets [i] based on digital (0, 11, 50)-net over F49, using
- digital (0, 5, 8)-net over F7, using
(16, 27, 177)-Net over F7 — Digital
Digital (16, 27, 177)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(727, 177, F7, 11) (dual of [177, 150, 12]-code), using
- construction XX applied to C1 = C([24,33]), C2 = C([23,32]), C3 = C1 + C2 = C([24,32]), and C∩ = C1 ∩ C2 = C([23,33]) [i] based on
- linear OA(724, 171, F7, 10) (dual of [171, 147, 11]-code), using the BCH-code C(I) with length 171 | 73−1, defining interval I = {24,25,…,33}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(724, 171, F7, 10) (dual of [171, 147, 11]-code), using the BCH-code C(I) with length 171 | 73−1, defining interval I = {23,24,…,32}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(727, 171, F7, 11) (dual of [171, 144, 12]-code), using the BCH-code C(I) with length 171 | 73−1, defining interval I = {23,24,…,33}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(721, 171, F7, 9) (dual of [171, 150, 10]-code), using the BCH-code C(I) with length 171 | 73−1, defining interval I = {24,25,…,32}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([24,33]), C2 = C([23,32]), C3 = C1 + C2 = C([24,32]), and C∩ = C1 ∩ C2 = C([23,33]) [i] based on
(16, 27, 10766)-Net in Base 7 — Upper bound on s
There is no (16, 27, 10767)-net in base 7, because
- 1 times m-reduction [i] would yield (16, 26, 10767)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 9389 011332 696294 645955 > 726 [i]