Best Known (172−16, 172, s)-Nets in Base 8
(172−16, 172, 2621440)-Net over F8 — Constructive and digital
Digital (156, 172, 2621440)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (42, 50, 524290)-net over F8, using
- net defined by OOA [i] based on linear OOA(850, 524290, F8, 8, 8) (dual of [(524290, 8), 4194270, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(850, 2097160, F8, 8) (dual of [2097160, 2097110, 9]-code), using
- construction X4 applied to C([1,8]) ⊂ C([1,6]) [i] based on
- linear OA(849, 2097151, F8, 8) (dual of [2097151, 2097102, 9]-code), using 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(842, 2097151, F8, 6) (dual of [2097151, 2097109, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(88, 9, F8, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,8)), using
- dual of repetition code with length 9 [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to C([1,8]) ⊂ C([1,6]) [i] based on
- OA 4-folding and stacking [i] based on linear OA(850, 2097160, F8, 8) (dual of [2097160, 2097110, 9]-code), using
- net defined by OOA [i] based on linear OOA(850, 524290, F8, 8, 8) (dual of [(524290, 8), 4194270, 9]-NRT-code), using
- digital (106, 122, 2097150)-net over F8, using
- net defined by OOA [i] based on linear OOA(8122, 2097150, F8, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8122, 8388601, F8, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8122, 8388602, F8, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
- trace code [i] based on linear OOA(6461, 4194301, F64, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6461, 8388602, F64, 16) (dual of [8388602, 8388541, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- OOA 2-folding [i] based on linear OA(6461, 8388602, F64, 16) (dual of [8388602, 8388541, 17]-code), using
- trace code [i] based on linear OOA(6461, 4194301, F64, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8122, 8388602, F8, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8122, 8388601, F8, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8122, 2097150, F8, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
- digital (42, 50, 524290)-net over F8, using
(172−16, 172, large)-Net over F8 — Digital
Digital (156, 172, large)-net over F8, using
- 84 times duplication [i] based on digital (152, 168, large)-net over F8, using
- t-expansion [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
- t-expansion [i] based on digital (144, 168, large)-net over F8, using
(172−16, 172, large)-Net in Base 8 — Upper bound on s
There is no (156, 172, large)-net in base 8, because
- 14 times m-reduction [i] would yield (156, 158, large)-net in base 8, but