Best Known (156, 178, s)-Nets in Base 3
(156, 178, 48316)-Net over F3 — Constructive and digital
Digital (156, 178, 48316)-net over F3, using
- 31 times duplication [i] based on digital (155, 177, 48316)-net over F3, using
- net defined by OOA [i] based on linear OOA(3177, 48316, F3, 22, 22) (dual of [(48316, 22), 1062775, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3177, 531476, F3, 22) (dual of [531476, 531299, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3177, 531482, F3, 22) (dual of [531482, 531305, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3177, 531482, F3, 22) (dual of [531482, 531305, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3177, 531476, F3, 22) (dual of [531476, 531299, 23]-code), using
- net defined by OOA [i] based on linear OOA(3177, 48316, F3, 22, 22) (dual of [(48316, 22), 1062775, 23]-NRT-code), using
(156, 178, 164336)-Net over F3 — Digital
Digital (156, 178, 164336)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3178, 164336, F3, 3, 22) (dual of [(164336, 3), 492830, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3178, 177162, F3, 3, 22) (dual of [(177162, 3), 531308, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3178, 531486, F3, 22) (dual of [531486, 531308, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(39, 45, F3, 4) (dual of [45, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(3178, 531486, F3, 22) (dual of [531486, 531308, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(3178, 177162, F3, 3, 22) (dual of [(177162, 3), 531308, 23]-NRT-code), using
(156, 178, large)-Net in Base 3 — Upper bound on s
There is no (156, 178, large)-net in base 3, because
- 20 times m-reduction [i] would yield (156, 158, large)-net in base 3, but