Best Known (70, 70+20, s)-Nets in Base 8
(70, 70+20, 3278)-Net over F8 — Constructive and digital
Digital (70, 90, 3278)-net over F8, using
- 82 times duplication [i] based on digital (68, 88, 3278)-net over F8, using
- net defined by OOA [i] based on linear OOA(888, 3278, F8, 20, 20) (dual of [(3278, 20), 65472, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(888, 32780, F8, 20) (dual of [32780, 32692, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(888, 32781, F8, 20) (dual of [32781, 32693, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- 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(876, 32768, F8, 18) (dual of [32768, 32692, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(871, 32768, F8, 17) (dual of [32768, 32697, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(19) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(888, 32781, F8, 20) (dual of [32781, 32693, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(888, 32780, F8, 20) (dual of [32780, 32692, 21]-code), using
- net defined by OOA [i] based on linear OOA(888, 3278, F8, 20, 20) (dual of [(3278, 20), 65472, 21]-NRT-code), using
(70, 70+20, 31489)-Net over F8 — Digital
Digital (70, 90, 31489)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(890, 31489, F8, 20) (dual of [31489, 31399, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(890, 32787, F8, 20) (dual of [32787, 32697, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(889, 32786, F8, 20) (dual of [32786, 32697, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- 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(871, 32768, F8, 17) (dual of [32768, 32697, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(19) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(889, 32786, F8, 20) (dual of [32786, 32697, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(890, 32787, F8, 20) (dual of [32787, 32697, 21]-code), using
(70, 70+20, large)-Net in Base 8 — Upper bound on s
There is no (70, 90, large)-net in base 8, because
- 18 times m-reduction [i] would yield (70, 72, large)-net in base 8, but