Best Known (69, 86, s)-Nets in Base 7
(69, 86, 14707)-Net over F7 — Constructive and digital
Digital (69, 86, 14707)-net over F7, using
- net defined by OOA [i] based on linear OOA(786, 14707, F7, 17, 17) (dual of [(14707, 17), 249933, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(786, 117657, F7, 17) (dual of [117657, 117571, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(786, 117663, F7, 17) (dual of [117663, 117577, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(785, 117650, F7, 17) (dual of [117650, 117565, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 13, F7, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(786, 117663, F7, 17) (dual of [117663, 117577, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(786, 117657, F7, 17) (dual of [117657, 117571, 18]-code), using
(69, 86, 65833)-Net over F7 — Digital
Digital (69, 86, 65833)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(786, 65833, F7, 17) (dual of [65833, 65747, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(786, 117663, F7, 17) (dual of [117663, 117577, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(785, 117650, F7, 17) (dual of [117650, 117565, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 13, F7, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(786, 117663, F7, 17) (dual of [117663, 117577, 18]-code), using
(69, 86, large)-Net in Base 7 — Upper bound on s
There is no (69, 86, large)-net in base 7, because
- 15 times m-reduction [i] would yield (69, 71, large)-net in base 7, but