Best Known (10, 10+6, s)-Nets in Base 9
(10, 10+6, 246)-Net over F9 — Constructive and digital
Digital (10, 16, 246)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 82)-net over F9, using
- net defined by OOA [i] based on linear OOA(94, 82, F9, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
- digital (6, 12, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 6, 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, 6, 82)-net over F81, using
- digital (1, 4, 82)-net over F9, using
(10, 10+6, 734)-Net over F9 — Digital
Digital (10, 16, 734)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(916, 734, F9, 6) (dual of [734, 718, 7]-code), using
- construction XX applied to C1 = C([727,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([727,4]) [i] based on
- linear OA(913, 728, F9, 5) (dual of [728, 715, 6]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,1,2,3}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(913, 728, F9, 5) (dual of [728, 715, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(916, 728, F9, 6) (dual of [728, 712, 7]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(910, 728, F9, 4) (dual of [728, 718, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [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([727,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([727,4]) [i] based on
(10, 10+6, 27897)-Net in Base 9 — Upper bound on s
There is no (10, 16, 27898)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 1853 113979 796145 > 916 [i]