Best Known (77−16, 77, s)-Nets in Base 7
(77−16, 77, 2115)-Net over F7 — Constructive and digital
Digital (61, 77, 2115)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 14)-net over F7, using
- 3 times m-reduction [i] based on digital (3, 14, 14)-net over F7, using
- digital (50, 66, 2101)-net over F7, using
- net defined by OOA [i] based on linear OOA(766, 2101, F7, 16, 16) (dual of [(2101, 16), 33550, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(766, 16808, F7, 16) (dual of [16808, 16742, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(766, 16812, F7, 16) (dual of [16812, 16746, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(766, 16807, F7, 16) (dual of [16807, 16741, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 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(766, 16812, F7, 16) (dual of [16812, 16746, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(766, 16808, F7, 16) (dual of [16808, 16742, 17]-code), using
- net defined by OOA [i] based on linear OOA(766, 2101, F7, 16, 16) (dual of [(2101, 16), 33550, 17]-NRT-code), using
- digital (3, 11, 14)-net over F7, using
(77−16, 77, 23330)-Net over F7 — Digital
Digital (61, 77, 23330)-net over F7, using
(77−16, 77, large)-Net in Base 7 — Upper bound on s
There is no (61, 77, large)-net in base 7, because
- 14 times m-reduction [i] would yield (61, 63, large)-net in base 7, but