Best Known (38, 38+8, s)-Nets in Base 7
(38, 38+8, 205890)-Net over F7 — Constructive and digital
Digital (38, 46, 205890)-net over F7, using
- net defined by OOA [i] based on linear OOA(746, 205890, F7, 8, 8) (dual of [(205890, 8), 1647074, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(746, 823560, F7, 8) (dual of [823560, 823514, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- 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(729, 823543, F7, 5) (dual of [823543, 823514, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(73, 17, F7, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- OA 4-folding and stacking [i] based on linear OA(746, 823560, F7, 8) (dual of [823560, 823514, 9]-code), using
(38, 38+8, 823560)-Net over F7 — Digital
Digital (38, 46, 823560)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(746, 823560, F7, 8) (dual of [823560, 823514, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- 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(729, 823543, F7, 5) (dual of [823543, 823514, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(73, 17, F7, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
(38, 38+8, large)-Net in Base 7 — Upper bound on s
There is no (38, 46, large)-net in base 7, because
- 6 times m-reduction [i] would yield (38, 40, large)-net in base 7, but