Best Known (84, 98, s)-Nets in Base 7
(84, 98, 823545)-Net over F7 — Constructive and digital
Digital (84, 98, 823545)-net over F7, using
- net defined by OOA [i] based on linear OOA(798, 823545, F7, 14, 14) (dual of [(823545, 14), 11529532, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(798, 5764815, F7, 14) (dual of [5764815, 5764717, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(798, 5764818, F7, 14) (dual of [5764818, 5764720, 15]-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(71, 17, F7, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(798, 5764818, F7, 14) (dual of [5764818, 5764720, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(798, 5764815, F7, 14) (dual of [5764815, 5764717, 15]-code), using
(84, 98, 5764819)-Net over F7 — Digital
Digital (84, 98, 5764819)-net over F7, using
- embedding of OOA with Gilbert–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
(84, 98, large)-Net in Base 7 — Upper bound on s
There is no (84, 98, large)-net in base 7, because
- 12 times m-reduction [i] would yield (84, 86, large)-net in base 7, but