Best Known (85, 85+19, s)-Nets in Base 7
(85, 85+19, 13076)-Net over F7 — Constructive and digital
Digital (85, 104, 13076)-net over F7, using
- net defined by OOA [i] based on linear OOA(7104, 13076, F7, 19, 19) (dual of [(13076, 19), 248340, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(7104, 117685, F7, 19) (dual of [117685, 117581, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(7104, 117686, F7, 19) (dual of [117686, 117582, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(797, 117649, F7, 19) (dual of [117649, 117552, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(77, 37, F7, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(7104, 117686, F7, 19) (dual of [117686, 117582, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(7104, 117685, F7, 19) (dual of [117685, 117581, 20]-code), using
(85, 85+19, 117686)-Net over F7 — Digital
Digital (85, 104, 117686)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7104, 117686, F7, 19) (dual of [117686, 117582, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(797, 117649, F7, 19) (dual of [117649, 117552, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(77, 37, F7, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
(85, 85+19, large)-Net in Base 7 — Upper bound on s
There is no (85, 104, large)-net in base 7, because
- 17 times m-reduction [i] would yield (85, 87, large)-net in base 7, but