Best Known (178−21, 178, s)-Nets in Base 3
(178−21, 178, 53149)-Net over F3 — Constructive and digital
Digital (157, 178, 53149)-net over F3, using
- net defined by OOA [i] based on linear OOA(3178, 53149, F3, 21, 21) (dual of [(53149, 21), 1115951, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3178, 531491, F3, 21) (dual of [531491, 531313, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3178, 531498, F3, 21) (dual of [531498, 531320, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [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(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(39, 57, F3, 4) (dual of [57, 48, 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(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3178, 531498, F3, 21) (dual of [531498, 531320, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3178, 531491, F3, 21) (dual of [531491, 531313, 22]-code), using
(178−21, 178, 177166)-Net over F3 — Digital
Digital (157, 178, 177166)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3178, 177166, F3, 3, 21) (dual of [(177166, 3), 531320, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3178, 531498, F3, 21) (dual of [531498, 531320, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [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(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(39, 57, F3, 4) (dual of [57, 48, 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(15) [i] based on
- OOA 3-folding [i] based on linear OA(3178, 531498, F3, 21) (dual of [531498, 531320, 22]-code), using
(178−21, 178, large)-Net in Base 3 — Upper bound on s
There is no (157, 178, large)-net in base 3, because
- 19 times m-reduction [i] would yield (157, 159, large)-net in base 3, but