Best Known (180, 180+26, s)-Nets in Base 3
(180, 180+26, 40881)-Net over F3 — Constructive and digital
Digital (180, 206, 40881)-net over F3, using
- 31 times duplication [i] based on digital (179, 205, 40881)-net over F3, using
- net defined by OOA [i] based on linear OOA(3205, 40881, F3, 26, 26) (dual of [(40881, 26), 1062701, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3205, 531453, F3, 26) (dual of [531453, 531248, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(30, 12, F3, 0) (dual of [12, 12, 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
- OA 13-folding and stacking [i] based on linear OA(3205, 531453, F3, 26) (dual of [531453, 531248, 27]-code), using
- net defined by OOA [i] based on linear OOA(3205, 40881, F3, 26, 26) (dual of [(40881, 26), 1062701, 27]-NRT-code), using
(180, 180+26, 132863)-Net over F3 — Digital
Digital (180, 206, 132863)-net over F3, using
- 31 times duplication [i] based on digital (179, 205, 132863)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3205, 132863, F3, 4, 26) (dual of [(132863, 4), 531247, 27]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3205, 531452, F3, 26) (dual of [531452, 531247, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3205, 531453, F3, 26) (dual of [531453, 531248, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(30, 12, F3, 0) (dual of [12, 12, 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(3205, 531453, F3, 26) (dual of [531453, 531248, 27]-code), using
- OOA 4-folding [i] based on linear OA(3205, 531452, F3, 26) (dual of [531452, 531247, 27]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3205, 132863, F3, 4, 26) (dual of [(132863, 4), 531247, 27]-NRT-code), using
(180, 180+26, large)-Net in Base 3 — Upper bound on s
There is no (180, 206, large)-net in base 3, because
- 24 times m-reduction [i] would yield (180, 182, large)-net in base 3, but