Best Known (144−32, 144, s)-Nets in Base 9
(144−32, 144, 3691)-Net over F9 — Constructive and digital
Digital (112, 144, 3691)-net over F9, using
- 92 times duplication [i] based on digital (110, 142, 3691)-net over F9, using
- net defined by OOA [i] based on linear OOA(9142, 3691, F9, 32, 32) (dual of [(3691, 32), 117970, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(9142, 59056, F9, 32) (dual of [59056, 58914, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(9142, 59060, F9, 32) (dual of [59060, 58918, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(29) [i] based on
- linear OA(9141, 59049, F9, 32) (dual of [59049, 58908, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(9131, 59049, F9, 30) (dual of [59049, 58918, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(91, 11, F9, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(31) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(9142, 59060, F9, 32) (dual of [59060, 58918, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(9142, 59056, F9, 32) (dual of [59056, 58914, 33]-code), using
- net defined by OOA [i] based on linear OOA(9142, 3691, F9, 32, 32) (dual of [(3691, 32), 117970, 33]-NRT-code), using
(144−32, 144, 53224)-Net over F9 — Digital
Digital (112, 144, 53224)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9144, 53224, F9, 32) (dual of [53224, 53080, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(9144, 59067, F9, 32) (dual of [59067, 58923, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- linear OA(9141, 59049, F9, 32) (dual of [59049, 58908, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(9126, 59049, F9, 29) (dual of [59049, 58923, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(93, 18, F9, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(9144, 59067, F9, 32) (dual of [59067, 58923, 33]-code), using
(144−32, 144, large)-Net in Base 9 — Upper bound on s
There is no (112, 144, large)-net in base 9, because
- 30 times m-reduction [i] would yield (112, 114, large)-net in base 9, but