Best Known (172, 172+23, s)-Nets in Base 3
(172, 172+23, 48323)-Net over F3 — Constructive and digital
Digital (172, 195, 48323)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- 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
- digital (3, 14, 10)-net over F3, using
(172, 172+23, 177167)-Net over F3 — Digital
Digital (172, 195, 177167)-net over F3, using
- 32 times duplication [i] based on digital (170, 193, 177167)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 177167, F3, 3, 23) (dual of [(177167, 3), 531308, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3193, 531501, F3, 23) (dual of [531501, 531308, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3192, 531500, F3, 23) (dual of [531500, 531308, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [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(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(311, 59, F3, 5) (dual of [59, 48, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3192, 531500, F3, 23) (dual of [531500, 531308, 24]-code), using
- OOA 3-folding [i] based on linear OA(3193, 531501, F3, 23) (dual of [531501, 531308, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 177167, F3, 3, 23) (dual of [(177167, 3), 531308, 24]-NRT-code), using
(172, 172+23, large)-Net in Base 3 — Upper bound on s
There is no (172, 195, large)-net in base 3, because
- 21 times m-reduction [i] would yield (172, 174, large)-net in base 3, but