Best Known (30, 30+10, s)-Nets in Base 8
(30, 30+10, 1640)-Net over F8 — Constructive and digital
Digital (30, 40, 1640)-net over F8, using
- net defined by OOA [i] based on linear OOA(840, 1640, F8, 10, 10) (dual of [(1640, 10), 16360, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(840, 8200, F8, 10) (dual of [8200, 8160, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(840, 8202, F8, 10) (dual of [8202, 8162, 11]-code), using
- trace code [i] based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(840, 8202, F8, 10) (dual of [8202, 8162, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(840, 8200, F8, 10) (dual of [8200, 8160, 11]-code), using
(30, 30+10, 8202)-Net over F8 — Digital
Digital (30, 40, 8202)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(840, 8202, F8, 10) (dual of [8202, 8162, 11]-code), using
- trace code [i] based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
(30, 30+10, 6243928)-Net in Base 8 — Upper bound on s
There is no (30, 40, 6243929)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 1 329228 603273 869978 820015 509225 430728 > 840 [i]