Best Known (44−10, 44, s)-Nets in Base 8
(44−10, 44, 6556)-Net over F8 — Constructive and digital
Digital (34, 44, 6556)-net over F8, using
- net defined by OOA [i] based on linear OOA(844, 6556, F8, 10, 10) (dual of [(6556, 10), 65516, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(844, 32780, F8, 10) (dual of [32780, 32736, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(844, 32781, F8, 10) (dual of [32781, 32737, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 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(831, 32768, F8, 7) (dual of [32768, 32737, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(83, 13, F8, 2) (dual of [13, 10, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(844, 32781, F8, 10) (dual of [32781, 32737, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(844, 32780, F8, 10) (dual of [32780, 32736, 11]-code), using
(44−10, 44, 32781)-Net over F8 — Digital
Digital (34, 44, 32781)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(844, 32781, F8, 10) (dual of [32781, 32737, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 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(831, 32768, F8, 7) (dual of [32768, 32737, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(83, 13, F8, 2) (dual of [13, 10, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
(44−10, 44, large)-Net in Base 8 — Upper bound on s
There is no (34, 44, large)-net in base 8, because
- 8 times m-reduction [i] would yield (34, 36, large)-net in base 8, but