Best Known (44, 44+8, s)-Nets in Base 7
(44, 44+8, 1441205)-Net over F7 — Constructive and digital
Digital (44, 52, 1441205)-net over F7, using
- net defined by OOA [i] based on linear OOA(752, 1441205, F7, 8, 8) (dual of [(1441205, 8), 11529588, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(752, 5764820, F7, 8) (dual of [5764820, 5764768, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [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(733, 5764801, F7, 5) (dual of [5764801, 5764768, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(73, 19, F7, 2) (dual of [19, 16, 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(752, 5764820, F7, 8) (dual of [5764820, 5764768, 9]-code), using
(44, 44+8, 5764820)-Net over F7 — Digital
Digital (44, 52, 5764820)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(752, 5764820, F7, 8) (dual of [5764820, 5764768, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [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(733, 5764801, F7, 5) (dual of [5764801, 5764768, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(73, 19, F7, 2) (dual of [19, 16, 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
(44, 44+8, large)-Net in Base 7 — Upper bound on s
There is no (44, 52, large)-net in base 7, because
- 6 times m-reduction [i] would yield (44, 46, large)-net in base 7, but