Best Known (239−22, 239, s)-Nets in Base 3
(239−22, 239, 762656)-Net over F3 — Constructive and digital
Digital (217, 239, 762656)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (17, 28, 56)-net over F3, using
- trace code for nets [i] based on digital (3, 14, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- trace code for nets [i] based on digital (3, 14, 28)-net over F9, 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 (17, 28, 56)-net over F3, using
(239−22, 239, 3545220)-Net over F3 — Digital
Digital (217, 239, 3545220)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3239, 3545220, F3, 2, 22) (dual of [(3545220, 2), 7090201, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3239, 4194357, F3, 2, 22) (dual of [(4194357, 2), 8388475, 23]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(328, 56, F3, 2, 11) (dual of [(56, 2), 84, 12]-NRT-code), using
- extracting embedded OOA [i] based on digital (17, 28, 56)-net over F3, using
- trace code for nets [i] based on digital (3, 14, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- trace code for nets [i] based on digital (3, 14, 28)-net over F9, using
- extracting embedded OOA [i] based on digital (17, 28, 56)-net over F3, 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(328, 56, F3, 2, 11) (dual of [(56, 2), 84, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(3239, 4194357, F3, 2, 22) (dual of [(4194357, 2), 8388475, 23]-NRT-code), using
(239−22, 239, large)-Net in Base 3 — Upper bound on s
There is no (217, 239, large)-net in base 3, because
- 20 times m-reduction [i] would yield (217, 219, large)-net in base 3, but