Best Known (40, 40+12, s)-Nets in Base 8
(40, 40+12, 5463)-Net over F8 — Constructive and digital
Digital (40, 52, 5463)-net over F8, using
- net defined by OOA [i] based on linear OOA(852, 5463, F8, 12, 12) (dual of [(5463, 12), 65504, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(852, 32778, F8, 12) (dual of [32778, 32726, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(852, 32779, F8, 12) (dual of [32779, 32727, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(852, 32779, F8, 12) (dual of [32779, 32727, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(852, 32778, F8, 12) (dual of [32778, 32726, 13]-code), using
(40, 40+12, 26094)-Net over F8 — Digital
Digital (40, 52, 26094)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(852, 26094, F8, 12) (dual of [26094, 26042, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(852, 32779, F8, 12) (dual of [32779, 32727, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(852, 32779, F8, 12) (dual of [32779, 32727, 13]-code), using
(40, 40+12, large)-Net in Base 8 — Upper bound on s
There is no (40, 52, large)-net in base 8, because
- 10 times m-reduction [i] would yield (40, 42, large)-net in base 8, but