Best Known (182−23, 182, s)-Nets in Base 3
(182−23, 182, 48313)-Net over F3 — Constructive and digital
Digital (159, 182, 48313)-net over F3, using
- 31 times duplication [i] based on digital (158, 181, 48313)-net over F3, using
- net defined by OOA [i] based on linear OOA(3181, 48313, F3, 23, 23) (dual of [(48313, 23), 1111018, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3181, 531444, F3, 23) (dual of [531444, 531263, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, 531453, F3, 23) (dual of [531453, 531272, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3181, 531453, F3, 23) (dual of [531453, 531272, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3181, 531444, F3, 23) (dual of [531444, 531263, 24]-code), using
- net defined by OOA [i] based on linear OOA(3181, 48313, F3, 23, 23) (dual of [(48313, 23), 1111018, 24]-NRT-code), using
(182−23, 182, 132863)-Net over F3 — Digital
Digital (159, 182, 132863)-net over F3, using
- 31 times duplication [i] based on digital (158, 181, 132863)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3181, 132863, F3, 4, 23) (dual of [(132863, 4), 531271, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3181, 531452, F3, 23) (dual of [531452, 531271, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, 531453, F3, 23) (dual of [531453, 531272, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3181, 531453, F3, 23) (dual of [531453, 531272, 24]-code), using
- OOA 4-folding [i] based on linear OA(3181, 531452, F3, 23) (dual of [531452, 531271, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3181, 132863, F3, 4, 23) (dual of [(132863, 4), 531271, 24]-NRT-code), using
(182−23, 182, large)-Net in Base 3 — Upper bound on s
There is no (159, 182, large)-net in base 3, because
- 21 times m-reduction [i] would yield (159, 161, large)-net in base 3, but