Best Known (55, 55+16, s)-Nets in Base 8
(55, 55+16, 4096)-Net over F8 — Constructive and digital
Digital (55, 71, 4096)-net over F8, using
- t-expansion [i] based on digital (54, 71, 4096)-net over F8, using
- net defined by OOA [i] based on linear OOA(871, 4096, F8, 17, 17) (dual of [(4096, 17), 69561, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(871, 32769, F8, 17) (dual of [32769, 32698, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(871, 32769, F8, 17) (dual of [32769, 32698, 18]-code), using
- net defined by OOA [i] based on linear OOA(871, 4096, F8, 17, 17) (dual of [(4096, 17), 69561, 18]-NRT-code), using
(55, 55+16, 28294)-Net over F8 — Digital
Digital (55, 71, 28294)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(871, 28294, F8, 16) (dual of [28294, 28223, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(871, 32773, F8, 16) (dual of [32773, 32702, 17]-code), using
- 1 times truncation [i] based on linear OA(872, 32774, F8, 17) (dual of [32774, 32702, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [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(866, 32768, F8, 15) (dual of [32768, 32702, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(81, 6, F8, 1) (dual of [6, 5, 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(14) [i] based on
- 1 times truncation [i] based on linear OA(872, 32774, F8, 17) (dual of [32774, 32702, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(871, 32773, F8, 16) (dual of [32773, 32702, 17]-code), using
(55, 55+16, large)-Net in Base 8 — Upper bound on s
There is no (55, 71, large)-net in base 8, because
- 14 times m-reduction [i] would yield (55, 57, large)-net in base 8, but