Best Known (30, 30+7, s)-Nets in Base 8
(30, 30+7, 87383)-Net over F8 — Constructive and digital
Digital (30, 37, 87383)-net over F8, using
- net defined by OOA [i] based on linear OOA(837, 87383, F8, 7, 7) (dual of [(87383, 7), 611644, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(837, 262150, F8, 7) (dual of [262150, 262113, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(837, 262144, F8, 7) (dual of [262144, 262107, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(831, 262144, F8, 6) (dual of [262144, 262113, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(837, 262150, F8, 7) (dual of [262150, 262113, 8]-code), using
(30, 30+7, 262150)-Net over F8 — Digital
Digital (30, 37, 262150)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(837, 262150, F8, 7) (dual of [262150, 262113, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(837, 262144, F8, 7) (dual of [262144, 262107, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(831, 262144, F8, 6) (dual of [262144, 262113, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 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(6) ⊂ Ce(5) [i] based on
(30, 30+7, large)-Net in Base 8 — Upper bound on s
There is no (30, 37, large)-net in base 8, because
- 5 times m-reduction [i] would yield (30, 32, large)-net in base 8, but