Best Known (65, 65+15, s)-Nets in Base 3
(65, 65+15, 937)-Net over F3 — Constructive and digital
Digital (65, 80, 937)-net over F3, using
- net defined by OOA [i] based on linear OOA(380, 937, F3, 15, 15) (dual of [(937, 15), 13975, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(380, 6560, F3, 15) (dual of [6560, 6480, 16]-code), using
- 1 times truncation [i] based on linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 1 times truncation [i] based on linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(380, 6560, F3, 15) (dual of [6560, 6480, 16]-code), using
(65, 65+15, 3280)-Net over F3 — Digital
Digital (65, 80, 3280)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(380, 3280, F3, 2, 15) (dual of [(3280, 2), 6480, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(380, 6560, F3, 15) (dual of [6560, 6480, 16]-code), using
- 1 times truncation [i] based on linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 1 times truncation [i] based on linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using
- OOA 2-folding [i] based on linear OA(380, 6560, F3, 15) (dual of [6560, 6480, 16]-code), using
(65, 65+15, 409766)-Net in Base 3 — Upper bound on s
There is no (65, 80, 409767)-net in base 3, because
- 1 times m-reduction [i] would yield (65, 79, 409767)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 49 270448 694611 665324 553459 780095 658259 > 379 [i]