Best Known (35, 35+12, s)-Nets in Base 8
(35, 35+12, 1366)-Net over F8 — Constructive and digital
Digital (35, 47, 1366)-net over F8, using
- 81 times duplication [i] based on digital (34, 46, 1366)-net over F8, using
- net defined by OOA [i] based on linear OOA(846, 1366, F8, 12, 12) (dual of [(1366, 12), 16346, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(846, 8196, F8, 12) (dual of [8196, 8150, 13]-code), using
- trace code [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- trace code [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(846, 8196, F8, 12) (dual of [8196, 8150, 13]-code), using
- net defined by OOA [i] based on linear OOA(846, 1366, F8, 12, 12) (dual of [(1366, 12), 16346, 13]-NRT-code), using
(35, 35+12, 8198)-Net over F8 — Digital
Digital (35, 47, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(847, 8198, F8, 12) (dual of [8198, 8151, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(846, 8196, F8, 12) (dual of [8196, 8150, 13]-code), using
- trace code [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- trace code [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
- linear OA(846, 8197, F8, 11) (dual of [8197, 8151, 12]-code), using Gilbert–Varšamov bound and bm = 846 > Vbs−1(k−1) = 105907 628336 112838 436641 451080 227184 342016 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 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
- linear OA(846, 8196, F8, 12) (dual of [8196, 8150, 13]-code), using
- construction X with Varšamov bound [i] based on
(35, 35+12, 5073745)-Net in Base 8 — Upper bound on s
There is no (35, 47, 5073746)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 2 787595 786782 025836 959695 727655 938557 432568 > 847 [i]