Best Known (83−19, 83, s)-Nets in Base 7
(83−19, 83, 1868)-Net over F7 — Constructive and digital
Digital (64, 83, 1868)-net over F7, using
- 71 times duplication [i] based on digital (63, 82, 1868)-net over F7, using
- net defined by OOA [i] based on linear OOA(782, 1868, F7, 19, 19) (dual of [(1868, 19), 35410, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(782, 16813, F7, 19) (dual of [16813, 16731, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(782, 16819, F7, 19) (dual of [16819, 16737, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(781, 16808, F7, 19) (dual of [16808, 16727, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(771, 16808, F7, 17) (dual of [16808, 16737, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 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([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(782, 16819, F7, 19) (dual of [16819, 16737, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(782, 16813, F7, 19) (dual of [16813, 16731, 20]-code), using
- net defined by OOA [i] based on linear OOA(782, 1868, F7, 19, 19) (dual of [(1868, 19), 35410, 20]-NRT-code), using
(83−19, 83, 14251)-Net over F7 — Digital
Digital (64, 83, 14251)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(783, 14251, F7, 19) (dual of [14251, 14168, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(783, 16820, F7, 19) (dual of [16820, 16737, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(782, 16819, F7, 19) (dual of [16819, 16737, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(781, 16808, F7, 19) (dual of [16808, 16727, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(771, 16808, F7, 17) (dual of [16808, 16737, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 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([0,9]) ⊂ C([0,8]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(782, 16819, F7, 19) (dual of [16819, 16737, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(783, 16820, F7, 19) (dual of [16820, 16737, 20]-code), using
(83−19, 83, large)-Net in Base 7 — Upper bound on s
There is no (64, 83, large)-net in base 7, because
- 17 times m-reduction [i] would yield (64, 66, large)-net in base 7, but