Best Known (26, 26+9, s)-Nets in Base 8
(26, 26+9, 2049)-Net over F8 — Constructive and digital
Digital (26, 35, 2049)-net over F8, using
- net defined by OOA [i] based on linear OOA(835, 2049, F8, 9, 9) (dual of [(2049, 9), 18406, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(835, 8197, F8, 9) (dual of [8197, 8162, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(834, 8196, F8, 9) (dual of [8196, 8162, 10]-code), using
- trace code [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(834, 8196, F8, 9) (dual of [8196, 8162, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(835, 8197, F8, 9) (dual of [8197, 8162, 10]-code), using
(26, 26+9, 8198)-Net over F8 — Digital
Digital (26, 35, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(835, 8198, F8, 9) (dual of [8198, 8163, 10]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(834, 8196, F8, 9) (dual of [8196, 8162, 10]-code), using
- trace code [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- linear OA(834, 8197, F8, 8) (dual of [8197, 8163, 9]-code), using Gilbert–Varšamov bound and bm = 834 > Vbs−1(k−1) = 404957 425498 565552 208393 889792 [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(834, 8196, F8, 9) (dual of [8196, 8162, 10]-code), using
- construction X with Varšamov bound [i] based on
(26, 26+9, large)-Net in Base 8 — Upper bound on s
There is no (26, 35, large)-net in base 8, because
- 7 times m-reduction [i] would yield (26, 28, large)-net in base 8, but