Best Known (61, 78, s)-Nets in Base 8
(61, 78, 4099)-Net over F8 — Constructive and digital
Digital (61, 78, 4099)-net over F8, using
- 81 times duplication [i] based on digital (60, 77, 4099)-net over F8, using
- net defined by OOA [i] based on linear OOA(877, 4099, F8, 17, 17) (dual of [(4099, 17), 69606, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(877, 32793, F8, 17) (dual of [32793, 32716, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(877, 32794, F8, 17) (dual of [32794, 32717, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- 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(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(877, 32794, F8, 17) (dual of [32794, 32717, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(877, 32793, F8, 17) (dual of [32793, 32716, 18]-code), using
- net defined by OOA [i] based on linear OOA(877, 4099, F8, 17, 17) (dual of [(4099, 17), 69606, 18]-NRT-code), using
(61, 78, 32800)-Net over F8 — Digital
Digital (61, 78, 32800)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(878, 32800, F8, 17) (dual of [32800, 32722, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- 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(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(87, 32, F8, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
(61, 78, large)-Net in Base 8 — Upper bound on s
There is no (61, 78, large)-net in base 8, because
- 15 times m-reduction [i] would yield (61, 63, large)-net in base 8, but