Best Known (178, 178+25, s)-Nets in Base 3
(178, 178+25, 44290)-Net over F3 — Constructive and digital
Digital (178, 203, 44290)-net over F3, using
- 32 times duplication [i] based on digital (176, 201, 44290)-net over F3, using
- net defined by OOA [i] based on linear OOA(3201, 44290, F3, 25, 25) (dual of [(44290, 25), 1107049, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3201, 531481, F3, 25) (dual of [531481, 531280, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3201, 531482, F3, 25) (dual of [531482, 531281, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3201, 531482, F3, 25) (dual of [531482, 531281, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3201, 531481, F3, 25) (dual of [531481, 531280, 26]-code), using
- net defined by OOA [i] based on linear OOA(3201, 44290, F3, 25, 25) (dual of [(44290, 25), 1107049, 26]-NRT-code), using
(178, 178+25, 151851)-Net over F3 — Digital
Digital (178, 203, 151851)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3203, 151851, F3, 3, 25) (dual of [(151851, 3), 455350, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3203, 177162, F3, 3, 25) (dual of [(177162, 3), 531283, 26]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3202, 177162, F3, 3, 25) (dual of [(177162, 3), 531284, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3202, 531486, F3, 25) (dual of [531486, 531284, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(24) ⊂ Ce(19) [i] based on
- OOA 3-folding [i] based on linear OA(3202, 531486, F3, 25) (dual of [531486, 531284, 26]-code), using
- 31 times duplication [i] based on linear OOA(3202, 177162, F3, 3, 25) (dual of [(177162, 3), 531284, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3203, 177162, F3, 3, 25) (dual of [(177162, 3), 531283, 26]-NRT-code), using
(178, 178+25, large)-Net in Base 3 — Upper bound on s
There is no (178, 203, large)-net in base 3, because
- 23 times m-reduction [i] would yield (178, 180, large)-net in base 3, but