Best Known (75, 75+16, s)-Nets in Base 3
(75, 75+16, 2460)-Net over F3 — Constructive and digital
Digital (75, 91, 2460)-net over F3, using
- net defined by OOA [i] based on linear OOA(391, 2460, F3, 16, 16) (dual of [(2460, 16), 39269, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(391, 19680, F3, 16) (dual of [19680, 19589, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(391, 19680, F3, 16) (dual of [19680, 19589, 17]-code), using
(75, 75+16, 6561)-Net over F3 — Digital
Digital (75, 91, 6561)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(391, 6561, F3, 3, 16) (dual of [(6561, 3), 19592, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OOA 3-folding [i] based on linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using
(75, 75+16, 503393)-Net in Base 3 — Upper bound on s
There is no (75, 91, 503394)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 26 183969 348327 674082 415976 755165 000303 504465 > 391 [i]