Best Known (85, 85+17, s)-Nets in Base 7
(85, 85+17, 102945)-Net over F7 — Constructive and digital
Digital (85, 102, 102945)-net over F7, using
- net defined by OOA [i] based on linear OOA(7102, 102945, F7, 17, 17) (dual of [(102945, 17), 1749963, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(7102, 823561, F7, 17) (dual of [823561, 823459, 18]-code), using
- 2 times code embedding in larger space [i] based on linear OA(7100, 823559, F7, 17) (dual of [823559, 823459, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(799, 823544, F7, 17) (dual of [823544, 823445, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 823544 | 714−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(785, 823544, F7, 15) (dual of [823544, 823459, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 823544 | 714−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 15, F7, 1) (dual of [15, 14, 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
- 2 times code embedding in larger space [i] based on linear OA(7100, 823559, F7, 17) (dual of [823559, 823459, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(7102, 823561, F7, 17) (dual of [823561, 823459, 18]-code), using
(85, 85+17, 524729)-Net over F7 — Digital
Digital (85, 102, 524729)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7102, 524729, F7, 17) (dual of [524729, 524627, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(7102, 823561, F7, 17) (dual of [823561, 823459, 18]-code), using
- 2 times code embedding in larger space [i] based on linear OA(7100, 823559, F7, 17) (dual of [823559, 823459, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(799, 823544, F7, 17) (dual of [823544, 823445, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 823544 | 714−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(785, 823544, F7, 15) (dual of [823544, 823459, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 823544 | 714−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 15, F7, 1) (dual of [15, 14, 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
- 2 times code embedding in larger space [i] based on linear OA(7100, 823559, F7, 17) (dual of [823559, 823459, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(7102, 823561, F7, 17) (dual of [823561, 823459, 18]-code), using
(85, 85+17, large)-Net in Base 7 — Upper bound on s
There is no (85, 102, large)-net in base 7, because
- 15 times m-reduction [i] would yield (85, 87, large)-net in base 7, but