Best Known (16, 16+8, s)-Nets in Base 9
(16, 16+8, 328)-Net over F9 — Constructive and digital
Digital (16, 24, 328)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (4, 8, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 4, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 4, 82)-net over F81, using
- digital (8, 16, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 8, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- trace code for nets [i] based on digital (0, 8, 82)-net over F81, using
- digital (4, 8, 164)-net over F9, using
(16, 16+8, 846)-Net over F9 — Digital
Digital (16, 24, 846)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(924, 846, F9, 8) (dual of [846, 822, 9]-code), using
- 110 step Varšamov–Edel lengthening with (ri) = (1, 15 times 0, 1, 93 times 0) [i] based on linear OA(922, 734, F9, 8) (dual of [734, 712, 9]-code), using
- construction XX applied to C1 = C([727,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([727,6]) [i] based on
- linear OA(919, 728, F9, 7) (dual of [728, 709, 8]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(919, 728, F9, 7) (dual of [728, 709, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(922, 728, F9, 8) (dual of [728, 706, 9]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(916, 728, F9, 6) (dual of [728, 712, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([727,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([727,6]) [i] based on
- 110 step Varšamov–Edel lengthening with (ri) = (1, 15 times 0, 1, 93 times 0) [i] based on linear OA(922, 734, F9, 8) (dual of [734, 712, 9]-code), using
(16, 16+8, 147032)-Net in Base 9 — Upper bound on s
There is no (16, 24, 147033)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 79768 533835 835576 015521 > 924 [i]