Best Known (67−15, 67, s)-Nets in Base 9
(67−15, 67, 8437)-Net over F9 — Constructive and digital
Digital (52, 67, 8437)-net over F9, using
- net defined by OOA [i] based on linear OOA(967, 8437, F9, 15, 15) (dual of [(8437, 15), 126488, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(967, 59060, F9, 15) (dual of [59060, 58993, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(966, 59049, F9, 15) (dual of [59049, 58983, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(956, 59049, F9, 13) (dual of [59049, 58993, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(967, 59060, F9, 15) (dual of [59060, 58993, 16]-code), using
(67−15, 67, 49530)-Net over F9 — Digital
Digital (52, 67, 49530)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(967, 49530, F9, 15) (dual of [49530, 49463, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(967, 59060, F9, 15) (dual of [59060, 58993, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(966, 59049, F9, 15) (dual of [59049, 58983, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(956, 59049, F9, 13) (dual of [59049, 58993, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(967, 59060, F9, 15) (dual of [59060, 58993, 16]-code), using
(67−15, 67, large)-Net in Base 9 — Upper bound on s
There is no (52, 67, large)-net in base 9, because
- 13 times m-reduction [i] would yield (52, 54, large)-net in base 9, but