Best Known (57, 71, s)-Nets in Base 8
(57, 71, 4705)-Net over F8 — Constructive and digital
Digital (57, 71, 4705)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 24)-net over F8, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- the Klein quartic over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- digital (47, 61, 4681)-net over F8, using
- net defined by OOA [i] based on linear OOA(861, 4681, F8, 14, 14) (dual of [(4681, 14), 65473, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(861, 32767, F8, 14) (dual of [32767, 32706, 15]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(861, 32767, F8, 14) (dual of [32767, 32706, 15]-code), using
- net defined by OOA [i] based on linear OOA(861, 4681, F8, 14, 14) (dual of [(4681, 14), 65473, 15]-NRT-code), using
- digital (3, 10, 24)-net over F8, using
(57, 71, 9362)-Net in Base 8 — Constructive
(57, 71, 9362)-net in base 8, using
- net defined by OOA [i] based on OOA(871, 9362, S8, 14, 14), using
- OA 7-folding and stacking [i] based on OA(871, 65534, S8, 14), using
- discarding factors based on OA(871, 65540, S8, 14), using
- discarding parts of the base [i] based on linear OA(1653, 65540, F16, 14) (dual of [65540, 65487, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(1653, 65536, F16, 14) (dual of [65536, 65483, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1649, 65536, F16, 13) (dual of [65536, 65487, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding parts of the base [i] based on linear OA(1653, 65540, F16, 14) (dual of [65540, 65487, 15]-code), using
- discarding factors based on OA(871, 65540, S8, 14), using
- OA 7-folding and stacking [i] based on OA(871, 65534, S8, 14), using
(57, 71, 69280)-Net over F8 — Digital
Digital (57, 71, 69280)-net over F8, using
(57, 71, large)-Net in Base 8 — Upper bound on s
There is no (57, 71, large)-net in base 8, because
- 12 times m-reduction [i] would yield (57, 59, large)-net in base 8, but