Best Known (42, 42+17, s)-Nets in Base 7
(42, 42+17, 301)-Net over F7 — Constructive and digital
Digital (42, 59, 301)-net over F7, using
- 71 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
(42, 42+17, 1974)-Net over F7 — Digital
Digital (42, 59, 1974)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(759, 1974, F7, 17) (dual of [1974, 1915, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(759, 2412, F7, 17) (dual of [2412, 2353, 18]-code), using
- 1 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
- 1 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(759, 2412, F7, 17) (dual of [2412, 2353, 18]-code), using
(42, 42+17, 840421)-Net in Base 7 — Upper bound on s
There is no (42, 59, 840422)-net in base 7, because
- 1 times m-reduction [i] would yield (42, 58, 840422)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 10 367851 335662 478189 572555 136573 474635 555904 570385 > 758 [i]