Best Known (56, 72, s)-Nets in Base 8
(56, 72, 4097)-Net over F8 — Constructive and digital
Digital (56, 72, 4097)-net over F8, using
- net defined by OOA [i] based on linear OOA(872, 4097, F8, 16, 16) (dual of [(4097, 16), 65480, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(872, 32776, F8, 16) (dual of [32776, 32704, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(872, 32779, F8, 16) (dual of [32779, 32707, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(871, 32768, F8, 17) (dual of [32768, 32697, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(872, 32779, F8, 16) (dual of [32779, 32707, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(872, 32776, F8, 16) (dual of [32776, 32704, 17]-code), using
(56, 72, 32779)-Net over F8 — Digital
Digital (56, 72, 32779)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(872, 32779, F8, 16) (dual of [32779, 32707, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(871, 32768, F8, 17) (dual of [32768, 32697, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
(56, 72, large)-Net in Base 8 — Upper bound on s
There is no (56, 72, large)-net in base 8, because
- 14 times m-reduction [i] would yield (56, 58, large)-net in base 8, but