Best Known (166−17, 166, s)-Nets in Base 8
(166−17, 166, 2105343)-Net over F8 — Constructive and digital
Digital (149, 166, 2105343)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (28, 36, 8193)-net over F8, using
- net defined by OOA [i] based on linear OOA(836, 8193, F8, 8, 8) (dual of [(8193, 8), 65508, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(836, 32772, F8, 8) (dual of [32772, 32736, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(836, 32773, F8, 8) (dual of [32773, 32737, 9]-code), using
- 1 times truncation [i] based on linear OA(837, 32774, F8, 9) (dual of [32774, 32737, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(831, 32768, F8, 7) (dual of [32768, 32737, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(81, 6, F8, 1) (dual of [6, 5, 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(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(837, 32774, F8, 9) (dual of [32774, 32737, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(836, 32773, F8, 8) (dual of [32773, 32737, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(836, 32772, F8, 8) (dual of [32772, 32736, 9]-code), using
- net defined by OOA [i] based on linear OOA(836, 8193, F8, 8, 8) (dual of [(8193, 8), 65508, 9]-NRT-code), using
- digital (113, 130, 2097150)-net over F8, using
- net defined by OOA [i] based on linear OOA(8130, 2097150, F8, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8130, 8388601, F8, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8130, 8388602, F8, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
- trace code [i] based on linear OOA(6465, 4194301, F64, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6465, 8388602, F64, 17) (dual of [8388602, 8388537, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- OOA 2-folding [i] based on linear OA(6465, 8388602, F64, 17) (dual of [8388602, 8388537, 18]-code), using
- trace code [i] based on linear OOA(6465, 4194301, F64, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8130, 8388602, F8, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8130, 8388601, F8, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8130, 2097150, F8, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
- digital (28, 36, 8193)-net over F8, using
(166−17, 166, large)-Net over F8 — Digital
Digital (149, 166, large)-net over F8, using
- t-expansion [i] based on digital (144, 166, large)-net over F8, using
- 2 times m-reduction [i] based on digital (144, 168, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8168, large, F8, 24) (dual of [large, large−168, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8168, large, F8, 24) (dual of [large, large−168, 25]-code), using
- 2 times m-reduction [i] based on digital (144, 168, large)-net over F8, using
(166−17, 166, large)-Net in Base 8 — Upper bound on s
There is no (149, 166, large)-net in base 8, because
- 15 times m-reduction [i] would yield (149, 151, large)-net in base 8, but