Best Known (85−19, 85, s)-Nets in Base 8
(85−19, 85, 3642)-Net over F8 — Constructive and digital
Digital (66, 85, 3642)-net over F8, using
- 83 times duplication [i] based on digital (63, 82, 3642)-net over F8, using
- net defined by OOA [i] based on linear OOA(882, 3642, F8, 19, 19) (dual of [(3642, 19), 69116, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(882, 32779, F8, 19) (dual of [32779, 32697, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(882, 32780, F8, 19) (dual of [32780, 32698, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(881, 32769, F8, 19) (dual of [32769, 32688, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 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]
- 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 C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(882, 32780, F8, 19) (dual of [32780, 32698, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(882, 32779, F8, 19) (dual of [32779, 32697, 20]-code), using
- net defined by OOA [i] based on linear OOA(882, 3642, F8, 19, 19) (dual of [(3642, 19), 69116, 20]-NRT-code), using
(85−19, 85, 29719)-Net over F8 — Digital
Digital (66, 85, 29719)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(885, 29719, F8, 19) (dual of [29719, 29634, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(885, 32787, F8, 19) (dual of [32787, 32702, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(881, 32768, F8, 19) (dual of [32768, 32687, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(84, 19, F8, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,8)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(885, 32787, F8, 19) (dual of [32787, 32702, 20]-code), using
(85−19, 85, large)-Net in Base 8 — Upper bound on s
There is no (66, 85, large)-net in base 8, because
- 17 times m-reduction [i] would yield (66, 68, large)-net in base 8, but