Best Known (23, 23+6, s)-Nets in Base 8
(23, 23+6, 10933)-Net over F8 — Constructive and digital
Digital (23, 29, 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 (20, 26, 10924)-net over F8, using
- net defined by OOA [i] based on linear OOA(826, 10924, F8, 6, 6) (dual of [(10924, 6), 65518, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(826, 32772, F8, 6) (dual of [32772, 32746, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(826, 32773, F8, 6) (dual of [32773, 32747, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- 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(821, 32768, F8, 5) (dual of [32768, 32747, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(826, 32773, F8, 6) (dual of [32773, 32747, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(826, 32772, F8, 6) (dual of [32772, 32746, 7]-code), using
- net defined by OOA [i] based on linear OOA(826, 10924, F8, 6, 6) (dual of [(10924, 6), 65518, 7]-NRT-code), using
- digital (0, 3, 9)-net over F8, using
(23, 23+6, 21847)-Net in Base 8 — Constructive
(23, 29, 21847)-net in base 8, using
- net defined by OOA [i] based on OOA(829, 21847, S8, 6, 6), using
- OA 3-folding and stacking [i] based on OA(829, 65541, S8, 6), using
- 1 times code embedding in larger space [i] based on OA(828, 65540, S8, 6), using
- discarding parts of the base [i] based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- 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(1617, 65536, F16, 5) (dual of [65536, 65519, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding parts of the base [i] based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- 1 times code embedding in larger space [i] based on OA(828, 65540, S8, 6), using
- OA 3-folding and stacking [i] based on OA(829, 65541, S8, 6), using
(23, 23+6, 64369)-Net over F8 — Digital
Digital (23, 29, 64369)-net over F8, using
(23, 23+6, 65540)-Net in Base 8
(23, 29, 65540)-net in base 8, using
- 81 times duplication [i] based on (22, 28, 65540)-net in base 8, using
- base change [i] based on digital (15, 21, 65540)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- 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(1617, 65536, F16, 5) (dual of [65536, 65519, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- base change [i] based on digital (15, 21, 65540)-net over F16, using
(23, 23+6, large)-Net in Base 8 — Upper bound on s
There is no (23, 29, large)-net in base 8, because
- 4 times m-reduction [i] would yield (23, 25, large)-net in base 8, but