Best Known (27, 27+7, s)-Nets in Base 8
(27, 27+7, 10933)-Net over F8 — Constructive and digital
Digital (27, 34, 10933)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (24, 31, 10924)-net over F8, using
- net defined by OOA [i] based on linear OOA(831, 10924, F8, 7, 7) (dual of [(10924, 7), 76437, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(831, 32773, F8, 7) (dual of [32773, 32742, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(831, 32768, F8, 7) (dual of [32768, 32737, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(826, 32768, F8, 6) (dual of [32768, 32742, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(80, 5, F8, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(831, 32773, F8, 7) (dual of [32773, 32742, 8]-code), using
- net defined by OOA [i] based on linear OOA(831, 10924, F8, 7, 7) (dual of [(10924, 7), 76437, 8]-NRT-code), using
- digital (0, 3, 9)-net over F8, using
(27, 27+7, 21846)-Net in Base 8 — Constructive
(27, 34, 21846)-net in base 8, using
- net defined by OOA [i] based on OOA(834, 21846, S8, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(834, 65539, S8, 7), using
- discarding factors based on OA(834, 65540, S8, 7), using
- discarding parts of the base [i] based on linear OA(1625, 65540, F16, 7) (dual of [65540, 65515, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(1625, 65536, F16, 7) (dual of [65536, 65511, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1621, 65536, F16, 6) (dual of [65536, 65515, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(160, 4, F16, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(1625, 65540, F16, 7) (dual of [65540, 65515, 8]-code), using
- discarding factors based on OA(834, 65540, S8, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(834, 65539, S8, 7), using
(27, 27+7, 56060)-Net over F8 — Digital
Digital (27, 34, 56060)-net over F8, using
(27, 27+7, large)-Net in Base 8 — Upper bound on s
There is no (27, 34, large)-net in base 8, because
- 5 times m-reduction [i] would yield (27, 29, large)-net in base 8, but