Best Known (211, 237, s)-Nets in Base 3
(211, 237, 122649)-Net over F3 — Constructive and digital
Digital (211, 237, 122649)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 15, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (196, 222, 122641)-net over F3, using
- net defined by OOA [i] based on linear OOA(3222, 122641, F3, 26, 26) (dual of [(122641, 26), 3188444, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3222, 1594333, F3, 26) (dual of [1594333, 1594111, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 1594336, F3, 26) (dual of [1594336, 1594114, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3222, 1594323, F3, 26) (dual of [1594323, 1594101, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 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(3222, 1594336, F3, 26) (dual of [1594336, 1594114, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3222, 1594333, F3, 26) (dual of [1594333, 1594111, 27]-code), using
- net defined by OOA [i] based on linear OOA(3222, 122641, F3, 26, 26) (dual of [(122641, 26), 3188444, 27]-NRT-code), using
- digital (2, 15, 8)-net over F3, using
(211, 237, 531467)-Net over F3 — Digital
Digital (211, 237, 531467)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3237, 531467, F3, 3, 26) (dual of [(531467, 3), 1594164, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3237, 1594401, F3, 26) (dual of [1594401, 1594164, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 1594403, F3, 26) (dual of [1594403, 1594166, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(3222, 1594323, F3, 26) (dual of [1594323, 1594101, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(315, 80, F3, 6) (dual of [80, 65, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3237, 1594403, F3, 26) (dual of [1594403, 1594166, 27]-code), using
- OOA 3-folding [i] based on linear OA(3237, 1594401, F3, 26) (dual of [1594401, 1594164, 27]-code), using
(211, 237, large)-Net in Base 3 — Upper bound on s
There is no (211, 237, large)-net in base 3, because
- 24 times m-reduction [i] would yield (211, 213, large)-net in base 3, but