Best Known (10−4, 10, s)-Nets in Base 9
(10−4, 10, 367)-Net over F9 — Constructive and digital
Digital (6, 10, 367)-net over F9, using
- net defined by OOA [i] based on linear OOA(910, 367, F9, 4, 4) (dual of [(367, 4), 1458, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(910, 367, F9, 3, 4) (dual of [(367, 3), 1091, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(910, 734, F9, 4) (dual of [734, 724, 5]-code), using
- construction XX applied to C1 = C([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- linear OA(97, 728, F9, 3) (dual of [728, 721, 4]-code or 728-cap in PG(6,9)), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(97, 728, F9, 3) (dual of [728, 721, 4]-code or 728-cap in PG(6,9)), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(910, 728, F9, 4) (dual of [728, 718, 5]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(94, 728, F9, 2) (dual of [728, 724, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [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([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- OA 2-folding and stacking [i] based on linear OA(910, 734, F9, 4) (dual of [734, 724, 5]-code), using
- appending kth column [i] based on linear OOA(910, 367, F9, 3, 4) (dual of [(367, 3), 1091, 5]-NRT-code), using
(10−4, 10, 734)-Net over F9 — Digital
Digital (6, 10, 734)-net over F9, using
- net defined by OOA [i] based on linear OOA(910, 734, F9, 4, 4) (dual of [(734, 4), 2926, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(910, 734, F9, 3, 4) (dual of [(734, 3), 2192, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(910, 734, F9, 4) (dual of [734, 724, 5]-code), using
- construction XX applied to C1 = C([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- linear OA(97, 728, F9, 3) (dual of [728, 721, 4]-code or 728-cap in PG(6,9)), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(97, 728, F9, 3) (dual of [728, 721, 4]-code or 728-cap in PG(6,9)), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(910, 728, F9, 4) (dual of [728, 718, 5]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(94, 728, F9, 2) (dual of [728, 724, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [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([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(910, 734, F9, 4) (dual of [734, 724, 5]-code), using
- appending kth column [i] based on linear OOA(910, 734, F9, 3, 4) (dual of [(734, 3), 2192, 5]-NRT-code), using
(10−4, 10, 10437)-Net in Base 9 — Upper bound on s
There is no (6, 10, 10438)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 3486 960033 > 910 [i]