Best Known (224, 224+22, s)-Nets in Base 3
(224, 224+22, 762684)-Net over F3 — Constructive and digital
Digital (224, 246, 762684)-net over F3, using
- 32 times duplication [i] based on digital (222, 244, 762684)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (22, 33, 84)-net over F3, using
- trace code for nets [i] based on digital (0, 11, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- trace code for nets [i] based on digital (0, 11, 28)-net over F27, 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 (22, 33, 84)-net over F3, using
- (u, u+v)-construction [i] based on
(224, 224+22, 4194426)-Net over F3 — Digital
Digital (224, 246, 4194426)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3246, 4194426, F3, 2, 22) (dual of [(4194426, 2), 8388606, 23]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(335, 125, F3, 2, 11) (dual of [(125, 2), 215, 12]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(335, 125, F3, 11) (dual of [125, 90, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(335, 126, F3, 11) (dual of [126, 91, 12]-code), using
- construction X applied to C({2,4,7,11,13,17,25}) ⊂ C({2,4,7,11,17,25}) [i] based on
- linear OA(335, 121, F3, 11) (dual of [121, 86, 12]-code), using the cyclic code C(A) with length 121 | 35−1, defining set A = {2,4,7,11,13,17,25}, and minimum distance d ≥ |{13,32,51,…,−39}|+1 = 12 (BCH-bound) [i]
- linear OA(330, 121, F3, 10) (dual of [121, 91, 11]-code), using the cyclic code C(A) with length 121 | 35−1, defining set A = {2,4,7,11,17,25}, and minimum distance d ≥ |{32,51,70,…,−39}|+1 = 11 (BCH-bound) [i]
- linear OA(30, 5, F3, 0) (dual of [5, 5, 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 C({2,4,7,11,13,17,25}) ⊂ C({2,4,7,11,17,25}) [i] based on
- discarding factors / shortening the dual code based on linear OA(335, 126, F3, 11) (dual of [126, 91, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(335, 125, F3, 11) (dual of [125, 90, 12]-code), using
- linear OOA(3211, 4194301, F3, 2, 22) (dual of [(4194301, 2), 8388391, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3211, 8388602, F3, 22) (dual of [8388602, 8388391, 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
- OOA 2-folding [i] based on linear OA(3211, 8388602, F3, 22) (dual of [8388602, 8388391, 23]-code), using
- linear OOA(335, 125, F3, 2, 11) (dual of [(125, 2), 215, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
(224, 224+22, large)-Net in Base 3 — Upper bound on s
There is no (224, 246, large)-net in base 3, because
- 20 times m-reduction [i] would yield (224, 226, large)-net in base 3, but