Best Known (42, 42+14, s)-Nets in Base 8
(42, 42+14, 1171)-Net over F8 — Constructive and digital
Digital (42, 56, 1171)-net over F8, using
- 81 times duplication [i] based on digital (41, 55, 1171)-net over F8, using
- net defined by OOA [i] based on linear OOA(855, 1171, F8, 14, 14) (dual of [(1171, 14), 16339, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(855, 8197, F8, 14) (dual of [8197, 8142, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(854, 8196, F8, 14) (dual of [8196, 8142, 15]-code), using
- trace code [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(854, 8196, F8, 14) (dual of [8196, 8142, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(855, 8197, F8, 14) (dual of [8197, 8142, 15]-code), using
- net defined by OOA [i] based on linear OOA(855, 1171, F8, 14, 14) (dual of [(1171, 14), 16339, 15]-NRT-code), using
(42, 42+14, 8202)-Net over F8 — Digital
Digital (42, 56, 8202)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(856, 8202, F8, 14) (dual of [8202, 8146, 15]-code), using
- trace code [i] based on linear OA(6428, 4101, F64, 14) (dual of [4101, 4073, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(6428, 4101, F64, 14) (dual of [4101, 4073, 15]-code), using
(42, 42+14, 8101030)-Net in Base 8 — Upper bound on s
There is no (42, 56, 8101031)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 374 144446 489199 740633 003392 011341 769335 104401 659800 > 856 [i]