Best Known (27−6, 27, s)-Nets in Base 9
(27−6, 27, 19687)-Net over F9 — Constructive and digital
Digital (21, 27, 19687)-net over F9, using
- net defined by OOA [i] based on linear OOA(927, 19687, F9, 6, 6) (dual of [(19687, 6), 118095, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(927, 59061, F9, 6) (dual of [59061, 59034, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(926, 59049, F9, 6) (dual of [59049, 59023, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(911, 12, F9, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,9)), using
- dual of repetition code with length 12 [i]
- linear OA(91, 12, F9, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(927, 59061, F9, 6) (dual of [59061, 59034, 7]-code), using
(27−6, 27, 59061)-Net over F9 — Digital
Digital (21, 27, 59061)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(927, 59061, F9, 6) (dual of [59061, 59034, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(926, 59049, F9, 6) (dual of [59049, 59023, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(911, 12, F9, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,9)), using
- dual of repetition code with length 12 [i]
- linear OA(91, 12, F9, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
(27−6, 27, large)-Net in Base 9 — Upper bound on s
There is no (21, 27, large)-net in base 9, because
- 4 times m-reduction [i] would yield (21, 23, large)-net in base 9, but