Best Known (165, 165+26, s)-Nets in Base 3
(165, 165+26, 13627)-Net over F3 — Constructive and digital
Digital (165, 191, 13627)-net over F3, using
- 33 times duplication [i] based on digital (162, 188, 13627)-net over F3, using
- net defined by OOA [i] based on linear OOA(3188, 13627, F3, 26, 26) (dual of [(13627, 26), 354114, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3188, 177151, F3, 26) (dual of [177151, 176963, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3188, 177158, F3, 26) (dual of [177158, 176970, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3188, 177147, F3, 26) (dual of [177147, 176959, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3188, 177158, F3, 26) (dual of [177158, 176970, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3188, 177151, F3, 26) (dual of [177151, 176963, 27]-code), using
- net defined by OOA [i] based on linear OOA(3188, 13627, F3, 26, 26) (dual of [(13627, 26), 354114, 27]-NRT-code), using
(165, 165+26, 54060)-Net over F3 — Digital
Digital (165, 191, 54060)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3191, 54060, F3, 3, 26) (dual of [(54060, 3), 161989, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3191, 59053, F3, 3, 26) (dual of [(59053, 3), 176968, 27]-NRT-code), using
- 32 times duplication [i] based on linear OOA(3189, 59053, F3, 3, 26) (dual of [(59053, 3), 176970, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3189, 177159, F3, 26) (dual of [177159, 176970, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3188, 177158, F3, 26) (dual of [177158, 176970, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3188, 177147, F3, 26) (dual of [177147, 176959, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3188, 177158, F3, 26) (dual of [177158, 176970, 27]-code), using
- OOA 3-folding [i] based on linear OA(3189, 177159, F3, 26) (dual of [177159, 176970, 27]-code), using
- 32 times duplication [i] based on linear OOA(3189, 59053, F3, 3, 26) (dual of [(59053, 3), 176970, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3191, 59053, F3, 3, 26) (dual of [(59053, 3), 176968, 27]-NRT-code), using
(165, 165+26, large)-Net in Base 3 — Upper bound on s
There is no (165, 191, large)-net in base 3, because
- 24 times m-reduction [i] would yield (165, 167, large)-net in base 3, but