Best Known (82−19, 82, s)-Nets in Base 8
(82−19, 82, 3642)-Net over F8 — Constructive and digital
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
(82−19, 82, 20587)-Net over F8 — Digital
Digital (63, 82, 20587)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(882, 20587, F8, 19) (dual of [20587, 20505, 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
(82−19, 82, large)-Net in Base 8 — Upper bound on s
There is no (63, 82, large)-net in base 8, because
- 17 times m-reduction [i] would yield (63, 65, large)-net in base 8, but