Best Known (87, 111, s)-Nets in Base 8
(87, 111, 2732)-Net over F8 — Constructive and digital
Digital (87, 111, 2732)-net over F8, using
- 81 times duplication [i] based on digital (86, 110, 2732)-net over F8, using
- t-expansion [i] based on digital (85, 110, 2732)-net over F8, using
- net defined by OOA [i] based on linear OOA(8110, 2732, F8, 25, 25) (dual of [(2732, 25), 68190, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8110, 32785, F8, 25) (dual of [32785, 32675, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8110, 32787, F8, 25) (dual of [32787, 32677, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- linear OA(8106, 32768, F8, 25) (dual of [32768, 32662, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(891, 32768, F8, 21) (dual of [32768, 32677, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(84, 19, F8, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,8)), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(8110, 32787, F8, 25) (dual of [32787, 32677, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8110, 32785, F8, 25) (dual of [32785, 32675, 26]-code), using
- net defined by OOA [i] based on linear OOA(8110, 2732, F8, 25, 25) (dual of [(2732, 25), 68190, 26]-NRT-code), using
- t-expansion [i] based on digital (85, 110, 2732)-net over F8, using
(87, 111, 32794)-Net over F8 — Digital
Digital (87, 111, 32794)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8111, 32794, F8, 24) (dual of [32794, 32683, 25]-code), using
- construction XX applied to Ce(24) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- linear OA(8106, 32768, F8, 25) (dual of [32768, 32662, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(84, 25, F8, 3) (dual of [25, 21, 4]-code or 25-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(24) ⊂ Ce(19) ⊂ Ce(18) [i] based on
(87, 111, large)-Net in Base 8 — Upper bound on s
There is no (87, 111, large)-net in base 8, because
- 22 times m-reduction [i] would yield (87, 89, large)-net in base 8, but