Best Known (58−10, 58, s)-Nets in Base 7
(58−10, 58, 164711)-Net over F7 — Constructive and digital
Digital (48, 58, 164711)-net over F7, using
- net defined by OOA [i] based on linear OOA(758, 164711, F7, 10, 10) (dual of [(164711, 10), 1647052, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(758, 823555, F7, 10) (dual of [823555, 823497, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(758, 823558, F7, 10) (dual of [823558, 823500, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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 Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(758, 823558, F7, 10) (dual of [823558, 823500, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(758, 823555, F7, 10) (dual of [823555, 823497, 11]-code), using
(58−10, 58, 658961)-Net over F7 — Digital
Digital (48, 58, 658961)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(758, 658961, F7, 10) (dual of [658961, 658903, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(758, 823558, F7, 10) (dual of [823558, 823500, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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 Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(758, 823558, F7, 10) (dual of [823558, 823500, 11]-code), using
(58−10, 58, large)-Net in Base 7 — Upper bound on s
There is no (48, 58, large)-net in base 7, because
- 8 times m-reduction [i] would yield (48, 50, large)-net in base 7, but