Best Known (60−12, 60, s)-Nets in Base 8
(60−12, 60, 5486)-Net over F8 — Constructive and digital
Digital (48, 60, 5486)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 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 (39, 51, 5462)-net over F8, using
- net defined by OOA [i] based on linear OOA(851, 5462, F8, 12, 12) (dual of [(5462, 12), 65493, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(851, 32772, F8, 12) (dual of [32772, 32721, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(851, 32773, F8, 12) (dual of [32773, 32722, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(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(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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(851, 32773, F8, 12) (dual of [32773, 32722, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(851, 32772, F8, 12) (dual of [32772, 32721, 13]-code), using
- net defined by OOA [i] based on linear OOA(851, 5462, F8, 12, 12) (dual of [(5462, 12), 65493, 13]-NRT-code), using
- digital (3, 9, 24)-net over F8, using
(60−12, 60, 10923)-Net in Base 8 — Constructive
(48, 60, 10923)-net in base 8, using
- base change [i] based on digital (33, 45, 10923)-net over F16, using
- net defined by OOA [i] based on linear OOA(1645, 10923, F16, 12, 12) (dual of [(10923, 12), 131031, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(1645, 65538, F16, 12) (dual of [65538, 65493, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1645, 65540, F16, 12) (dual of [65540, 65495, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(1645, 65536, F16, 12) (dual of [65536, 65491, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1641, 65536, F16, 11) (dual of [65536, 65495, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(1645, 65540, F16, 12) (dual of [65540, 65495, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(1645, 65538, F16, 12) (dual of [65538, 65493, 13]-code), using
- net defined by OOA [i] based on linear OOA(1645, 10923, F16, 12, 12) (dual of [(10923, 12), 131031, 13]-NRT-code), using
(60−12, 60, 59143)-Net over F8 — Digital
Digital (48, 60, 59143)-net over F8, using
(60−12, 60, 59976)-Net in Base 8
(48, 60, 59976)-net in base 8, using
- base change [i] based on digital (33, 45, 59976)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1645, 59976, F16, 12) (dual of [59976, 59931, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1645, 65536, F16, 12) (dual of [65536, 65491, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(1645, 65536, F16, 12) (dual of [65536, 65491, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1645, 59976, F16, 12) (dual of [59976, 59931, 13]-code), using
(60−12, 60, large)-Net in Base 8 — Upper bound on s
There is no (48, 60, large)-net in base 8, because
- 10 times m-reduction [i] would yield (48, 50, large)-net in base 8, but