Best Known (227−22, 227, s)-Nets in Base 3
(227−22, 227, 762613)-Net over F3 — Constructive and digital
Digital (205, 227, 762613)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 16, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- digital (189, 211, 762600)-net over F3, using
- net defined by OOA [i] based on linear OOA(3211, 762600, F3, 22, 22) (dual of [(762600, 22), 16776989, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3211, 8388600, F3, 22) (dual of [8388600, 8388389, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, large, F3, 22) (dual of [large, large−211, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(3211, large, F3, 22) (dual of [large, large−211, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3211, 8388600, F3, 22) (dual of [8388600, 8388389, 23]-code), using
- net defined by OOA [i] based on linear OOA(3211, 762600, F3, 22, 22) (dual of [(762600, 22), 16776989, 23]-NRT-code), using
- digital (5, 16, 13)-net over F3, using
(227−22, 227, 2796214)-Net over F3 — Digital
Digital (205, 227, 2796214)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3227, 2796214, F3, 3, 22) (dual of [(2796214, 3), 8388415, 23]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(316, 13, F3, 3, 11) (dual of [(13, 3), 23, 12]-NRT-code), using
- extracting embedded OOA [i] based on digital (5, 16, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- extracting embedded OOA [i] based on digital (5, 16, 13)-net over F3, using
- linear OOA(3211, 2796201, F3, 3, 22) (dual of [(2796201, 3), 8388392, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3211, large, F3, 22) (dual of [large, large−211, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- OOA 3-folding [i] based on linear OA(3211, large, F3, 22) (dual of [large, large−211, 23]-code), using
- linear OOA(316, 13, F3, 3, 11) (dual of [(13, 3), 23, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
(227−22, 227, large)-Net in Base 3 — Upper bound on s
There is no (205, 227, large)-net in base 3, because
- 20 times m-reduction [i] would yield (205, 207, large)-net in base 3, but