Best Known (58, 58+16, s)-Nets in Base 9
(58, 58+16, 7383)-Net over F9 — Constructive and digital
Digital (58, 74, 7383)-net over F9, using
- net defined by OOA [i] based on linear OOA(974, 7383, F9, 16, 16) (dual of [(7383, 16), 118054, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(974, 59064, F9, 16) (dual of [59064, 58990, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(974, 59067, F9, 16) (dual of [59067, 58993, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(971, 59049, F9, 16) (dual of [59049, 58978, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(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(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(974, 59067, F9, 16) (dual of [59067, 58993, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(974, 59064, F9, 16) (dual of [59064, 58990, 17]-code), using
(58, 58+16, 59067)-Net over F9 — Digital
Digital (58, 74, 59067)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(974, 59067, F9, 16) (dual of [59067, 58993, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(971, 59049, F9, 16) (dual of [59049, 58978, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(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(15) ⊂ Ce(12) [i] based on
(58, 58+16, large)-Net in Base 9 — Upper bound on s
There is no (58, 74, large)-net in base 9, because
- 14 times m-reduction [i] would yield (58, 60, large)-net in base 9, but