Best Known (57, 73, s)-Nets in Base 9
(57, 73, 7382)-Net over F9 — Constructive and digital
Digital (57, 73, 7382)-net over F9, using
- 91 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(972, 7382, F9, 16, 16) (dual of [(7382, 16), 118040, 17]-NRT-code), using
(57, 73, 59062)-Net over F9 — Digital
Digital (57, 73, 59062)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(973, 59062, F9, 16) (dual of [59062, 58989, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ 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(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(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, 12, F9, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [i] based on
(57, 73, large)-Net in Base 9 — Upper bound on s
There is no (57, 73, large)-net in base 9, because
- 14 times m-reduction [i] would yield (57, 59, large)-net in base 9, but