Best Known (56, 72, s)-Nets in Base 9
(56, 72, 7382)-Net over F9 — Constructive and digital
Digital (56, 72, 7382)-net over F9, using
- net defined by OOA [i] based on linear OOA(972, 7382, F9, 16, 16) (dual of [(7382, 16), 118040, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(972, 59056, F9, 16) (dual of [59056, 58984, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(972, 59060, F9, 16) (dual of [59060, 58988, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(961, 59049, F9, 14) (dual of [59049, 58988, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(972, 59060, F9, 16) (dual of [59060, 58988, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(972, 59056, F9, 16) (dual of [59056, 58984, 17]-code), using
(56, 72, 52201)-Net over F9 — Digital
Digital (56, 72, 52201)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(972, 52201, F9, 16) (dual of [52201, 52129, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(972, 59060, F9, 16) (dual of [59060, 58988, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(961, 59049, F9, 14) (dual of [59049, 58988, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(972, 59060, F9, 16) (dual of [59060, 58988, 17]-code), using
(56, 72, large)-Net in Base 9 — Upper bound on s
There is no (56, 72, large)-net in base 9, because
- 14 times m-reduction [i] would yield (56, 58, large)-net in base 9, but