Best Known (135, 135+23, s)-Nets in Base 8
(135, 135+23, 190679)-Net over F8 — Constructive and digital
Digital (135, 158, 190679)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (5, 16, 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 (119, 142, 190651)-net over F8, using
- net defined by OOA [i] based on linear OOA(8142, 190651, F8, 23, 23) (dual of [(190651, 23), 4384831, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8142, 2097162, F8, 23) (dual of [2097162, 2097020, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8142, 2097168, F8, 23) (dual of [2097168, 2097026, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8141, 2097153, F8, 23) (dual of [2097153, 2097012, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8127, 2097153, F8, 21) (dual of [2097153, 2097026, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(81, 15, F8, 1) (dual of [15, 14, 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 C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8142, 2097168, F8, 23) (dual of [2097168, 2097026, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8142, 2097162, F8, 23) (dual of [2097162, 2097020, 24]-code), using
- net defined by OOA [i] based on linear OOA(8142, 190651, F8, 23, 23) (dual of [(190651, 23), 4384831, 24]-NRT-code), using
- digital (5, 16, 28)-net over F8, using
(135, 135+23, 381300)-Net in Base 8 — Constructive
(135, 158, 381300)-net in base 8, using
- net defined by OOA [i] based on OOA(8158, 381300, S8, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(8158, 4194301, S8, 23), using
- discarding factors based on OA(8158, 4194310, S8, 23), using
- trace code [i] based on OA(6479, 2097155, S64, 23), using
- discarding parts of the base [i] based on linear OA(12867, 2097155, F128, 23) (dual of [2097155, 2097088, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 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(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- discarding parts of the base [i] based on linear OA(12867, 2097155, F128, 23) (dual of [2097155, 2097088, 24]-code), using
- trace code [i] based on OA(6479, 2097155, S64, 23), using
- discarding factors based on OA(8158, 4194310, S8, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(8158, 4194301, S8, 23), using
(135, 135+23, 3958989)-Net over F8 — Digital
Digital (135, 158, 3958989)-net over F8, using
(135, 135+23, large)-Net in Base 8 — Upper bound on s
There is no (135, 158, large)-net in base 8, because
- 21 times m-reduction [i] would yield (135, 137, large)-net in base 8, but