Best Known (42, 42+8, s)-Nets in Base 7
(42, 42+8, 1441202)-Net over F7 — Constructive and digital
Digital (42, 50, 1441202)-net over F7, using
- net defined by OOA [i] based on linear OOA(750, 1441202, F7, 8, 8) (dual of [(1441202, 8), 11529566, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(750, 5764808, F7, 8) (dual of [5764808, 5764758, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(750, 5764810, F7, 8) (dual of [5764810, 5764760, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(741, 5764801, F7, 6) (dual of [5764801, 5764760, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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 Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(750, 5764810, F7, 8) (dual of [5764810, 5764760, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(750, 5764808, F7, 8) (dual of [5764808, 5764758, 9]-code), using
(42, 42+8, 3978364)-Net over F7 — Digital
Digital (42, 50, 3978364)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(750, 3978364, F7, 8) (dual of [3978364, 3978314, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(750, 5764810, F7, 8) (dual of [5764810, 5764760, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(741, 5764801, F7, 6) (dual of [5764801, 5764760, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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 Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(750, 5764810, F7, 8) (dual of [5764810, 5764760, 9]-code), using
(42, 42+8, large)-Net in Base 7 — Upper bound on s
There is no (42, 50, large)-net in base 7, because
- 6 times m-reduction [i] would yield (42, 44, large)-net in base 7, but