Best Known (87−19, 87, s)-Nets in Base 8
(87−19, 87, 3643)-Net over F8 — Constructive and digital
Digital (68, 87, 3643)-net over F8, using
- 81 times duplication [i] based on digital (67, 86, 3643)-net over F8, using
- net defined by OOA [i] based on linear OOA(886, 3643, F8, 19, 19) (dual of [(3643, 19), 69131, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(886, 32788, F8, 19) (dual of [32788, 32702, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(886, 32789, F8, 19) (dual of [32789, 32703, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- 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(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(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(84, 20, F8, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,8)), 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(18) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(886, 32789, F8, 19) (dual of [32789, 32703, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(886, 32788, F8, 19) (dual of [32788, 32702, 20]-code), using
- net defined by OOA [i] based on linear OOA(886, 3643, F8, 19, 19) (dual of [(3643, 19), 69131, 20]-NRT-code), using
(87−19, 87, 32794)-Net over F8 — Digital
Digital (68, 87, 32794)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(887, 32794, F8, 19) (dual of [32794, 32707, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- 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(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(86, 26, F8, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
(87−19, 87, large)-Net in Base 8 — Upper bound on s
There is no (68, 87, large)-net in base 8, because
- 17 times m-reduction [i] would yield (68, 70, large)-net in base 8, but