Best Known (72, 72+16, s)-Nets in Base 7
(72, 72+16, 14719)-Net over F7 — Constructive and digital
Digital (72, 88, 14719)-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 (63, 79, 14706)-net over F7, using
- net defined by OOA [i] based on linear OOA(779, 14706, F7, 16, 16) (dual of [(14706, 16), 235217, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(779, 117648, F7, 16) (dual of [117648, 117569, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(779, 117649, F7, 16) (dual of [117649, 117570, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(779, 117649, F7, 16) (dual of [117649, 117570, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(779, 117648, F7, 16) (dual of [117648, 117569, 17]-code), using
- net defined by OOA [i] based on linear OOA(779, 14706, F7, 16, 16) (dual of [(14706, 16), 235217, 17]-NRT-code), using
- digital (1, 9, 13)-net over F7, using
(72, 72+16, 117690)-Net over F7 — Digital
Digital (72, 88, 117690)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(788, 117690, F7, 16) (dual of [117690, 117602, 17]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(786, 117686, F7, 16) (dual of [117686, 117600, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- linear OA(779, 117649, F7, 16) (dual of [117649, 117570, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(749, 117649, F7, 10) (dual of [117649, 117600, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(15) ⊂ Ce(9) [i] based on
- linear OA(786, 117688, F7, 15) (dual of [117688, 117602, 16]-code), using Gilbert–Varšamov bound and bm = 786 > Vbs−1(k−1) = 87817 451724 636967 389692 410859 285688 681074 312109 236557 365406 878924 357751 [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(786, 117686, F7, 16) (dual of [117686, 117600, 17]-code), using
- construction X with Varšamov bound [i] based on
(72, 72+16, large)-Net in Base 7 — Upper bound on s
There is no (72, 88, large)-net in base 7, because
- 14 times m-reduction [i] would yield (72, 74, large)-net in base 7, but