Best Known (69−15, 69, s)-Nets in Base 8
(69−15, 69, 4683)-Net over F8 — Constructive and digital
Digital (54, 69, 4683)-net over F8, using
- net defined by OOA [i] based on linear OOA(869, 4683, F8, 15, 15) (dual of [(4683, 15), 70176, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(869, 32782, F8, 15) (dual of [32782, 32713, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(869, 32786, F8, 15) (dual of [32786, 32717, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(866, 32768, F8, 15) (dual of [32768, 32702, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(83, 18, F8, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(869, 32786, F8, 15) (dual of [32786, 32717, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(869, 32782, F8, 15) (dual of [32782, 32713, 16]-code), using
(69−15, 69, 32786)-Net over F8 — Digital
Digital (54, 69, 32786)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(869, 32786, F8, 15) (dual of [32786, 32717, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(866, 32768, F8, 15) (dual of [32768, 32702, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(83, 18, F8, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
(69−15, 69, large)-Net in Base 8 — Upper bound on s
There is no (54, 69, large)-net in base 8, because
- 13 times m-reduction [i] would yield (54, 56, large)-net in base 8, but