Best Known (85, 99, s)-Nets in Base 7
(85, 99, 823545)-Net over F7 — Constructive and digital
Digital (85, 99, 823545)-net over F7, using
- 1 times m-reduction [i] based on digital (85, 100, 823545)-net over F7, using
- net defined by OOA [i] based on linear OOA(7100, 823545, F7, 15, 15) (dual of [(823545, 15), 12353075, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(7100, 5764816, F7, 15) (dual of [5764816, 5764716, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(7100, 5764820, F7, 15) (dual of [5764820, 5764720, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(797, 5764801, F7, 15) (dual of [5764801, 5764704, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(7100, 5764820, F7, 15) (dual of [5764820, 5764720, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(7100, 5764816, F7, 15) (dual of [5764816, 5764716, 16]-code), using
- net defined by OOA [i] based on linear OOA(7100, 823545, F7, 15, 15) (dual of [(823545, 15), 12353075, 16]-NRT-code), using
(85, 99, 5764821)-Net over F7 — Digital
Digital (85, 99, 5764821)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(799, 5764821, F7, 14) (dual of [5764821, 5764722, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(798, 5764819, F7, 14) (dual of [5764819, 5764721, 15]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(797, 5764801, F7, 15) (dual of [5764801, 5764704, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(717, 18, F7, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,7)), using
- dual of repetition code with length 18 [i]
- linear OA(71, 18, F7, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(798, 5764820, F7, 13) (dual of [5764820, 5764722, 14]-code), using Gilbert–Varšamov bound and bm = 798 > Vbs−1(k−1) = 6122 112879 854602 069341 865726 028346 402677 503102 377336 240096 469420 914502 801220 494807 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 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
- linear OA(798, 5764819, F7, 14) (dual of [5764819, 5764721, 15]-code), using
- construction X with Varšamov bound [i] based on
(85, 99, large)-Net in Base 7 — Upper bound on s
There is no (85, 99, large)-net in base 7, because
- 12 times m-reduction [i] would yield (85, 87, large)-net in base 7, but