Best Known (85, 101, s)-Nets in Base 7
(85, 101, 102956)-Net over F7 — Constructive and digital
Digital (85, 101, 102956)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 13)-net over F7, using
- 4 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (76, 92, 102943)-net over F7, using
- net defined by OOA [i] based on linear OOA(792, 102943, F7, 16, 16) (dual of [(102943, 16), 1646996, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(792, 823544, F7, 16) (dual of [823544, 823452, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(792, 823550, F7, 16) (dual of [823550, 823458, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(792, 823543, F7, 16) (dual of [823543, 823451, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(15) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(792, 823550, F7, 16) (dual of [823550, 823458, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(792, 823544, F7, 16) (dual of [823544, 823452, 17]-code), using
- net defined by OOA [i] based on linear OOA(792, 102943, F7, 16, 16) (dual of [(102943, 16), 1646996, 17]-NRT-code), using
- digital (1, 9, 13)-net over F7, using
(85, 101, 823589)-Net over F7 — Digital
Digital (85, 101, 823589)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7101, 823589, F7, 16) (dual of [823589, 823488, 17]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(799, 823585, F7, 16) (dual of [823585, 823486, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- linear OA(792, 823543, F7, 16) (dual of [823543, 823451, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(77, 42, F7, 5) (dual of [42, 35, 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(15) ⊂ Ce(9) [i] based on
- linear OA(799, 823587, F7, 15) (dual of [823587, 823488, 16]-code), using Gilbert–Varšamov bound and bm = 799 > Vbs−1(k−1) = 59372 833329 019777 294765 131087 128305 971941 415829 571332 254287 374598 811751 373425 484593 [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 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(799, 823585, F7, 16) (dual of [823585, 823486, 17]-code), using
- construction X with Varšamov bound [i] based on
(85, 101, large)-Net in Base 7 — Upper bound on s
There is no (85, 101, large)-net in base 7, because
- 14 times m-reduction [i] would yield (85, 87, large)-net in base 7, but