Best Known (43, 43+17, s)-Nets in Base 7
(43, 43+17, 301)-Net over F7 — Constructive and digital
Digital (43, 60, 301)-net over F7, using
- 72 times duplication [i] based on digital (41, 58, 301)-net over F7, using
- net defined by OOA [i] based on linear OOA(758, 301, F7, 17, 17) (dual of [(301, 17), 5059, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(758, 2409, F7, 17) (dual of [2409, 2351, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(758, 2411, F7, 17) (dual of [2411, 2353, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(757, 2402, F7, 17) (dual of [2402, 2345, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(749, 2402, F7, 15) (dual of [2402, 2353, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 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(758, 2411, F7, 17) (dual of [2411, 2353, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(758, 2409, F7, 17) (dual of [2409, 2351, 18]-code), using
- net defined by OOA [i] based on linear OOA(758, 301, F7, 17, 17) (dual of [(301, 17), 5059, 18]-NRT-code), using
(43, 43+17, 2249)-Net over F7 — Digital
Digital (43, 60, 2249)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(760, 2249, F7, 17) (dual of [2249, 2189, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(760, 2413, F7, 17) (dual of [2413, 2353, 18]-code), using
- 2 times code embedding in larger space [i] based on linear OA(758, 2411, F7, 17) (dual of [2411, 2353, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(757, 2402, F7, 17) (dual of [2402, 2345, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(749, 2402, F7, 15) (dual of [2402, 2353, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 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(758, 2411, F7, 17) (dual of [2411, 2353, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(760, 2413, F7, 17) (dual of [2413, 2353, 18]-code), using
(43, 43+17, 1071852)-Net in Base 7 — Upper bound on s
There is no (43, 60, 1071853)-net in base 7, because
- 1 times m-reduction [i] would yield (43, 59, 1071853)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 72 574987 420651 189423 730009 159593 206811 935816 314881 > 759 [i]