Best Known (9, 9+6, s)-Nets in Base 16
(9, 9+6, 531)-Net over F16 — Constructive and digital
Digital (9, 15, 531)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (6, 12, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 6, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 6, 257)-net over F256, using
- digital (0, 3, 17)-net over F16, using
(9, 9+6, 825)-Net over F16 — Digital
Digital (9, 15, 825)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1615, 825, F16, 6) (dual of [825, 810, 7]-code), using
- construction XX applied to C1 = C([106,110]), C2 = C([105,109]), C3 = C1 + C2 = C([106,109]), and C∩ = C1 ∩ C2 = C([105,110]) [i] based on
- linear OA(1612, 819, F16, 5) (dual of [819, 807, 6]-code), using the BCH-code C(I) with length 819 | 163−1, defining interval I = {106,107,108,109,110}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1612, 819, F16, 5) (dual of [819, 807, 6]-code), using the BCH-code C(I) with length 819 | 163−1, defining interval I = {105,106,107,108,109}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1615, 819, F16, 6) (dual of [819, 804, 7]-code), using the BCH-code C(I) with length 819 | 163−1, defining interval I = {105,106,…,110}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(169, 819, F16, 4) (dual of [819, 810, 5]-code), using the BCH-code C(I) with length 819 | 163−1, defining interval I = {106,107,108,109}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(160, 3, F16, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([106,110]), C2 = C([105,109]), C3 = C1 + C2 = C([106,109]), and C∩ = C1 ∩ C2 = C([105,110]) [i] based on
(9, 9+6, 983)-Net in Base 16
(9, 15, 983)-net in base 16, using
- base change [i] based on digital (6, 12, 983)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3212, 983, F32, 6) (dual of [983, 971, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(3212, 1029, F32, 6) (dual of [1029, 1017, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(3211, 1024, F32, 6) (dual of [1024, 1013, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(327, 1024, F32, 4) (dual of [1024, 1017, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(3212, 1029, F32, 6) (dual of [1029, 1017, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3212, 983, F32, 6) (dual of [983, 971, 7]-code), using
(9, 9+6, 127024)-Net in Base 16 — Upper bound on s
There is no (9, 15, 127025)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 1 152928 687918 107376 > 1615 [i]