Best Known (72, 72+18, s)-Nets in Base 8
(72, 72+18, 3669)-Net over F8 — Constructive and digital
Digital (72, 90, 3669)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (5, 14, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- digital (58, 76, 3641)-net over F8, using
- net defined by OOA [i] based on linear OOA(876, 3641, F8, 18, 18) (dual of [(3641, 18), 65462, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(876, 32769, F8, 18) (dual of [32769, 32693, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(876, 32773, F8, 18) (dual of [32773, 32697, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(876, 32768, F8, 18) (dual of [32768, 32692, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(876, 32773, F8, 18) (dual of [32773, 32697, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(876, 32769, F8, 18) (dual of [32769, 32693, 19]-code), using
- net defined by OOA [i] based on linear OOA(876, 3641, F8, 18, 18) (dual of [(3641, 18), 65462, 19]-NRT-code), using
- digital (5, 14, 28)-net over F8, using
(72, 72+18, 7282)-Net in Base 8 — Constructive
(72, 90, 7282)-net in base 8, using
- 82 times duplication [i] based on (70, 88, 7282)-net in base 8, using
- base change [i] based on digital (48, 66, 7282)-net over F16, using
- 161 times duplication [i] based on digital (47, 65, 7282)-net over F16, using
- net defined by OOA [i] based on linear OOA(1665, 7282, F16, 18, 18) (dual of [(7282, 18), 131011, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(1665, 65538, F16, 18) (dual of [65538, 65473, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1665, 65540, F16, 18) (dual of [65540, 65475, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(1665, 65536, F16, 18) (dual of [65536, 65471, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(1665, 65540, F16, 18) (dual of [65540, 65475, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(1665, 65538, F16, 18) (dual of [65538, 65473, 19]-code), using
- net defined by OOA [i] based on linear OOA(1665, 7282, F16, 18, 18) (dual of [(7282, 18), 131011, 19]-NRT-code), using
- 161 times duplication [i] based on digital (47, 65, 7282)-net over F16, using
- base change [i] based on digital (48, 66, 7282)-net over F16, using
(72, 72+18, 61940)-Net over F8 — Digital
Digital (72, 90, 61940)-net over F8, using
(72, 72+18, large)-Net in Base 8 — Upper bound on s
There is no (72, 90, large)-net in base 8, because
- 16 times m-reduction [i] would yield (72, 74, large)-net in base 8, but