Best Known (29, 29+12, s)-Nets in Base 8
(29, 29+12, 683)-Net over F8 — Constructive and digital
Digital (29, 41, 683)-net over F8, using
- net defined by OOA [i] based on linear OOA(841, 683, F8, 12, 12) (dual of [(683, 12), 8155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(841, 4098, F8, 12) (dual of [4098, 4057, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(841, 4100, F8, 12) (dual of [4100, 4059, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(841, 4096, F8, 12) (dual of [4096, 4055, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(837, 4096, F8, 11) (dual of [4096, 4059, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(841, 4100, F8, 12) (dual of [4100, 4059, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(841, 4098, F8, 12) (dual of [4098, 4057, 13]-code), using
(29, 29+12, 2645)-Net over F8 — Digital
Digital (29, 41, 2645)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(841, 2645, F8, 12) (dual of [2645, 2604, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(841, 4096, F8, 12) (dual of [4096, 4055, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(841, 4096, F8, 12) (dual of [4096, 4055, 13]-code), using
(29, 29+12, 634215)-Net in Base 8 — Upper bound on s
There is no (29, 41, 634216)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 10 633903 188239 392579 425092 608763 342927 > 841 [i]