Best Known (37, 37+12, s)-Nets in Base 8
(37, 37+12, 1367)-Net over F8 — Constructive and digital
Digital (37, 49, 1367)-net over F8, using
- 81 times duplication [i] based on digital (36, 48, 1367)-net over F8, using
- net defined by OOA [i] based on linear OOA(848, 1367, F8, 12, 12) (dual of [(1367, 12), 16356, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(848, 8202, F8, 12) (dual of [8202, 8154, 13]-code), using
- trace code [i] based on linear OA(6424, 4101, F64, 12) (dual of [4101, 4077, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [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(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(6424, 4101, F64, 12) (dual of [4101, 4077, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(848, 8202, F8, 12) (dual of [8202, 8154, 13]-code), using
- net defined by OOA [i] based on linear OOA(848, 1367, F8, 12, 12) (dual of [(1367, 12), 16356, 13]-NRT-code), using
(37, 37+12, 8204)-Net over F8 — Digital
Digital (37, 49, 8204)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(849, 8204, F8, 12) (dual of [8204, 8155, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(848, 8202, F8, 12) (dual of [8202, 8154, 13]-code), using
- trace code [i] based on linear OA(6424, 4101, F64, 12) (dual of [4101, 4077, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [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(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(6424, 4101, F64, 12) (dual of [4101, 4077, 13]-code), using
- linear OA(848, 8203, F8, 11) (dual of [8203, 8155, 12]-code), using Gilbert–Varšamov bound and bm = 848 > Vbs−1(k−1) = 106685 914692 012626 189131 416557 594022 498304 [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(848, 8202, F8, 12) (dual of [8202, 8154, 13]-code), using
- construction X with Varšamov bound [i] based on
(37, 37+12, large)-Net in Base 8 — Upper bound on s
There is no (37, 49, large)-net in base 8, because
- 10 times m-reduction [i] would yield (37, 39, large)-net in base 8, but