Best Known (89, 89+10, s)-Nets in Base 8
(89, 89+10, 3486514)-Net over F8 — Constructive and digital
Digital (89, 99, 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 (64, 74, 3355440)-net over F8, using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F64, using
- digital (20, 25, 131074)-net over F8, using
(89, 89+10, large)-Net over F8 — Digital
Digital (89, 99, large)-net over F8, using
- 82 times duplication [i] based on digital (87, 97, large)-net over F8, using
- t-expansion [i] based on digital (83, 97, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(897, large, F8, 14) (dual of [large, large−97, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(897, large, F8, 14) (dual of [large, large−97, 15]-code), using
- t-expansion [i] based on digital (83, 97, large)-net over F8, using
(89, 89+10, large)-Net in Base 8 — Upper bound on s
There is no (89, 99, large)-net in base 8, because
- 8 times m-reduction [i] would yield (89, 91, large)-net in base 8, but