Best Known (52−8, 52, s)-Nets in Base 8
(52−8, 52, 524292)-Net over F8 — Constructive and digital
Digital (44, 52, 524292)-net over F8, using
- 1 times m-reduction [i] based on digital (44, 53, 524292)-net over F8, using
- net defined by OOA [i] based on linear OOA(853, 524292, F8, 9, 9) (dual of [(524292, 9), 4718575, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(853, 2097169, F8, 9) (dual of [2097169, 2097116, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(850, 2097152, F8, 9) (dual of [2097152, 2097102, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(83, 17, F8, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(853, 2097169, F8, 9) (dual of [2097169, 2097116, 10]-code), using
- net defined by OOA [i] based on linear OOA(853, 524292, F8, 9, 9) (dual of [(524292, 9), 4718575, 10]-NRT-code), using
(52−8, 52, 1048577)-Net in Base 8 — Constructive
(44, 52, 1048577)-net in base 8, using
- net defined by OOA [i] based on OOA(852, 1048577, S8, 8, 8), using
- OA 4-folding and stacking [i] based on OA(852, 4194308, S8, 8), using
- discarding factors based on OA(852, 4194310, S8, 8), using
- trace code [i] based on OA(6426, 2097155, S64, 8), using
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- trace code [i] based on OA(6426, 2097155, S64, 8), using
- discarding factors based on OA(852, 4194310, S8, 8), using
- OA 4-folding and stacking [i] based on OA(852, 4194308, S8, 8), using
(52−8, 52, 2468821)-Net over F8 — Digital
Digital (44, 52, 2468821)-net over F8, using
(52−8, 52, large)-Net in Base 8 — Upper bound on s
There is no (44, 52, large)-net in base 8, because
- 6 times m-reduction [i] would yield (44, 46, large)-net in base 8, but