Best Known (67, 67+19, s)-Nets in Base 7
(67, 67+19, 1870)-Net over F7 — Constructive and digital
Digital (67, 86, 1870)-net over F7, using
- 71 times duplication [i] based on digital (66, 85, 1870)-net over F7, using
- net defined by OOA [i] based on linear OOA(785, 1870, F7, 19, 19) (dual of [(1870, 19), 35445, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(785, 16831, F7, 19) (dual of [16831, 16746, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(785, 16832, F7, 19) (dual of [16832, 16747, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [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(761, 16808, F7, 15) (dual of [16808, 16747, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(74, 24, F7, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,7)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(785, 16832, F7, 19) (dual of [16832, 16747, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(785, 16831, F7, 19) (dual of [16831, 16746, 20]-code), using
- net defined by OOA [i] based on linear OOA(785, 1870, F7, 19, 19) (dual of [(1870, 19), 35445, 20]-NRT-code), using
(67, 67+19, 16834)-Net over F7 — Digital
Digital (67, 86, 16834)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(786, 16834, F7, 19) (dual of [16834, 16748, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(785, 16832, F7, 19) (dual of [16832, 16747, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [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(761, 16808, F7, 15) (dual of [16808, 16747, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(74, 24, F7, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,7)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(785, 16833, F7, 18) (dual of [16833, 16748, 19]-code), using Gilbert–Varšamov bound and bm = 785 > Vbs−1(k−1) = 32989 294105 206223 001405 174883 080732 242307 607219 744237 045160 634340 751873 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(785, 16832, F7, 19) (dual of [16832, 16747, 20]-code), using
- construction X with Varšamov bound [i] based on
(67, 67+19, large)-Net in Base 7 — Upper bound on s
There is no (67, 86, large)-net in base 7, because
- 17 times m-reduction [i] would yield (67, 69, large)-net in base 7, but