Best Known (107−11, 107, s)-Nets in Base 8
(107−11, 107, 3486514)-Net over F8 — Constructive and digital
Digital (96, 107, 3486514)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (20, 25, 131074)-net over F8, using
- net defined by OOA [i] based on linear OOA(825, 131074, F8, 5, 5) (dual of [(131074, 5), 655345, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(825, 262149, F8, 5) (dual of [262149, 262124, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(825, 262150, F8, 5) (dual of [262150, 262125, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(825, 262144, F8, 5) (dual of [262144, 262119, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(819, 262144, F8, 4) (dual of [262144, 262125, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(825, 262150, F8, 5) (dual of [262150, 262125, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(825, 262149, F8, 5) (dual of [262149, 262124, 6]-code), using
- net defined by OOA [i] based on linear OOA(825, 131074, F8, 5, 5) (dual of [(131074, 5), 655345, 6]-NRT-code), using
- digital (71, 82, 3355440)-net over F8, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- digital (20, 25, 131074)-net over F8, using
(107−11, 107, large)-Net over F8 — Digital
Digital (96, 107, large)-net over F8, using
- 5 times m-reduction [i] based on digital (96, 112, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8112, large, F8, 16) (dual of [large, large−112, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8112, large, F8, 16) (dual of [large, large−112, 17]-code), using
(107−11, 107, large)-Net in Base 8 — Upper bound on s
There is no (96, 107, large)-net in base 8, because
- 9 times m-reduction [i] would yield (96, 98, large)-net in base 8, but