Best Known (20, 20+7, s)-Nets in Base 8
(20, 20+7, 2732)-Net over F8 — Constructive and digital
Digital (20, 27, 2732)-net over F8, using
- net defined by OOA [i] based on linear OOA(827, 2732, F8, 7, 7) (dual of [(2732, 7), 19097, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(827, 8197, F8, 7) (dual of [8197, 8170, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(826, 8196, F8, 7) (dual of [8196, 8170, 8]-code), using
- trace code [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(826, 8196, F8, 7) (dual of [8196, 8170, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(827, 8197, F8, 7) (dual of [8197, 8170, 8]-code), using
(20, 20+7, 8198)-Net over F8 — Digital
Digital (20, 27, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(827, 8198, F8, 7) (dual of [8198, 8171, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(826, 8196, F8, 7) (dual of [8196, 8170, 8]-code), using
- trace code [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- linear OA(826, 8197, F8, 6) (dual of [8197, 8171, 7]-code), using Gilbert–Varšamov bound and bm = 826 > Vbs−1(k−1) = 5174 001952 051112 933376 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(826, 8196, F8, 7) (dual of [8196, 8170, 8]-code), using
- construction X with Varšamov bound [i] based on
(20, 20+7, large)-Net in Base 8 — Upper bound on s
There is no (20, 27, large)-net in base 8, because
- 5 times m-reduction [i] would yield (20, 22, large)-net in base 8, but