Best Known (55, 65, s)-Nets in Base 7
(55, 65, 1152961)-Net over F7 — Constructive and digital
Digital (55, 65, 1152961)-net over F7, using
- net defined by OOA [i] based on linear OOA(765, 1152961, F7, 10, 10) (dual of [(1152961, 10), 11529545, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(765, 5764805, F7, 10) (dual of [5764805, 5764740, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(765, 5764809, F7, 10) (dual of [5764809, 5764744, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(757, 5764801, F7, 9) (dual of [5764801, 5764744, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(765, 5764809, F7, 10) (dual of [5764809, 5764744, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(765, 5764805, F7, 10) (dual of [5764805, 5764740, 11]-code), using
(55, 65, 3616784)-Net over F7 — Digital
Digital (55, 65, 3616784)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(765, 3616784, F7, 10) (dual of [3616784, 3616719, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using
(55, 65, large)-Net in Base 7 — Upper bound on s
There is no (55, 65, large)-net in base 7, because
- 8 times m-reduction [i] would yield (55, 57, large)-net in base 7, but