Best Known (77, 77+22, s)-Nets in Base 8
(77, 77+22, 2980)-Net over F8 — Constructive and digital
Digital (77, 99, 2980)-net over F8, using
- 81 times duplication [i] based on digital (76, 98, 2980)-net over F8, using
- net defined by OOA [i] based on linear OOA(898, 2980, F8, 22, 22) (dual of [(2980, 22), 65462, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(898, 32780, F8, 22) (dual of [32780, 32682, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(898, 32781, F8, 22) (dual of [32781, 32683, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- linear OA(896, 32768, F8, 22) (dual of [32768, 32672, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(886, 32768, F8, 20) (dual of [32768, 32682, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(881, 32768, F8, 19) (dual of [32768, 32687, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(81, 12, F8, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(21) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(898, 32781, F8, 22) (dual of [32781, 32683, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(898, 32780, F8, 22) (dual of [32780, 32682, 23]-code), using
- net defined by OOA [i] based on linear OOA(898, 2980, F8, 22, 22) (dual of [(2980, 22), 65462, 23]-NRT-code), using
(77, 77+22, 31564)-Net over F8 — Digital
Digital (77, 99, 31564)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(899, 31564, F8, 22) (dual of [31564, 31465, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(899, 32786, F8, 22) (dual of [32786, 32687, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(896, 32768, F8, 22) (dual of [32768, 32672, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(881, 32768, F8, 19) (dual of [32768, 32687, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(899, 32786, F8, 22) (dual of [32786, 32687, 23]-code), using
(77, 77+22, large)-Net in Base 8 — Upper bound on s
There is no (77, 99, large)-net in base 8, because
- 20 times m-reduction [i] would yield (77, 79, large)-net in base 8, but