Best Known (135−29, 135, s)-Nets in Base 9
(135−29, 135, 4220)-Net over F9 — Constructive and digital
Digital (106, 135, 4220)-net over F9, using
- 92 times duplication [i] based on digital (104, 133, 4220)-net over F9, using
- net defined by OOA [i] based on linear OOA(9133, 4220, F9, 29, 29) (dual of [(4220, 29), 122247, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(9133, 59081, F9, 29) (dual of [59081, 58948, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- 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(9101, 59049, F9, 23) (dual of [59049, 58948, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(97, 32, F9, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(9133, 59081, F9, 29) (dual of [59081, 58948, 30]-code), using
- net defined by OOA [i] based on linear OOA(9133, 4220, F9, 29, 29) (dual of [(4220, 29), 122247, 30]-NRT-code), using
(135−29, 135, 59088)-Net over F9 — Digital
Digital (106, 135, 59088)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9135, 59088, F9, 29) (dual of [59088, 58953, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- 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(996, 59049, F9, 22) (dual of [59049, 58953, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(99, 39, F9, 6) (dual of [39, 30, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(99, 41, F9, 6) (dual of [41, 32, 7]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(99, 41, F9, 6) (dual of [41, 32, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
(135−29, 135, large)-Net in Base 9 — Upper bound on s
There is no (106, 135, large)-net in base 9, because
- 27 times m-reduction [i] would yield (106, 108, large)-net in base 9, but