Best Known (30, 30+6, s)-Nets in Base 7
(30, 30+6, 274516)-Net over F7 — Constructive and digital
Digital (30, 36, 274516)-net over F7, using
- net defined by OOA [i] based on linear OOA(736, 274516, F7, 6, 6) (dual of [(274516, 6), 1647060, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(736, 823548, F7, 6) (dual of [823548, 823512, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(736, 823550, F7, 6) (dual of [823550, 823514, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(736, 823543, F7, 6) (dual of [823543, 823507, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(70, 7, F7, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(736, 823550, F7, 6) (dual of [823550, 823514, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(736, 823548, F7, 6) (dual of [823548, 823512, 7]-code), using
(30, 30+6, 823550)-Net over F7 — Digital
Digital (30, 36, 823550)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(736, 823550, F7, 6) (dual of [823550, 823514, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(736, 823543, F7, 6) (dual of [823543, 823507, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(70, 7, F7, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
(30, 30+6, large)-Net in Base 7 — Upper bound on s
There is no (30, 36, large)-net in base 7, because
- 4 times m-reduction [i] would yield (30, 32, large)-net in base 7, but