Best Known (68, 81, s)-Nets in Base 8
(68, 81, 349529)-Net over F8 — Constructive and digital
Digital (68, 81, 349529)-net over F8, using
- net defined by OOA [i] based on linear OOA(881, 349529, F8, 13, 13) (dual of [(349529, 13), 4543796, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(881, 2097175, F8, 13) (dual of [2097175, 2097094, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(881, 2097176, F8, 13) (dual of [2097176, 2097095, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(878, 2097152, F8, 13) (dual of [2097152, 2097074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(83, 24, F8, 2) (dual of [24, 21, 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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(881, 2097176, F8, 13) (dual of [2097176, 2097095, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(881, 2097175, F8, 13) (dual of [2097175, 2097094, 14]-code), using
(68, 81, 2097176)-Net over F8 — Digital
Digital (68, 81, 2097176)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(881, 2097176, F8, 13) (dual of [2097176, 2097095, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(878, 2097152, F8, 13) (dual of [2097152, 2097074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(83, 24, F8, 2) (dual of [24, 21, 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(12) ⊂ Ce(9) [i] based on
(68, 81, large)-Net in Base 8 — Upper bound on s
There is no (68, 81, large)-net in base 8, because
- 11 times m-reduction [i] would yield (68, 70, large)-net in base 8, but