Best Known (165−24, 165, s)-Nets in Base 8
(165−24, 165, 174791)-Net over F8 — Constructive and digital
Digital (141, 165, 174791)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (5, 17, 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 (124, 148, 174763)-net over F8, using
- net defined by OOA [i] based on linear OOA(8148, 174763, F8, 24, 24) (dual of [(174763, 24), 4194164, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(8148, 2097156, F8, 24) (dual of [2097156, 2097008, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(8148, 2097159, F8, 24) (dual of [2097159, 2097011, 25]-code), using
- 1 times truncation [i] based on linear OA(8149, 2097160, F8, 25) (dual of [2097160, 2097011, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8141, 2097152, F8, 23) (dual of [2097152, 2097011, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(81, 8, F8, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(8149, 2097160, F8, 25) (dual of [2097160, 2097011, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8148, 2097159, F8, 24) (dual of [2097159, 2097011, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(8148, 2097156, F8, 24) (dual of [2097156, 2097008, 25]-code), using
- net defined by OOA [i] based on linear OOA(8148, 174763, F8, 24, 24) (dual of [(174763, 24), 4194164, 25]-NRT-code), using
- digital (5, 17, 28)-net over F8, using
(165−24, 165, 349525)-Net in Base 8 — Constructive
(141, 165, 349525)-net in base 8, using
- 81 times duplication [i] based on (140, 164, 349525)-net in base 8, using
- net defined by OOA [i] based on OOA(8164, 349525, S8, 24, 24), using
- OA 12-folding and stacking [i] based on OA(8164, 4194300, S8, 24), using
- discarding factors based on OA(8164, 4194310, S8, 24), using
- trace code [i] based on OA(6482, 2097155, S64, 24), using
- discarding parts of the base [i] based on linear OA(12870, 2097155, F128, 24) (dual of [2097155, 2097085, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12867, 2097152, F128, 23) (dual of [2097152, 2097085, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- discarding parts of the base [i] based on linear OA(12870, 2097155, F128, 24) (dual of [2097155, 2097085, 25]-code), using
- trace code [i] based on OA(6482, 2097155, S64, 24), using
- discarding factors based on OA(8164, 4194310, S8, 24), using
- OA 12-folding and stacking [i] based on OA(8164, 4194300, S8, 24), using
- net defined by OOA [i] based on OOA(8164, 349525, S8, 24, 24), using
(165−24, 165, 4055538)-Net over F8 — Digital
Digital (141, 165, 4055538)-net over F8, using
(165−24, 165, large)-Net in Base 8 — Upper bound on s
There is no (141, 165, large)-net in base 8, because
- 22 times m-reduction [i] would yield (141, 143, large)-net in base 8, but