Best Known (243−22, 243, s)-Nets in Base 3
(243−22, 243, 762664)-Net over F3 — Constructive and digital
Digital (221, 243, 762664)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (21, 32, 64)-net over F3, using
- trace code for nets [i] based on digital (5, 16, 32)-net over F9, using
- net from sequence [i] based on digital (5, 31)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 5 and N(F) ≥ 32, using
- net from sequence [i] based on digital (5, 31)-sequence over F9, using
- trace code for nets [i] based on digital (5, 16, 32)-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 (21, 32, 64)-net over F3, using
(243−22, 243, 4194386)-Net over F3 — Digital
Digital (221, 243, 4194386)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3243, 4194386, F3, 2, 22) (dual of [(4194386, 2), 8388529, 23]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(332, 85, F3, 2, 11) (dual of [(85, 2), 138, 12]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(332, 85, F3, 11) (dual of [85, 53, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(332, 87, F3, 11) (dual of [87, 55, 12]-code), using
- (u, u+v)-construction [i] based on
- linear OA(35, 6, F3, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,3)), using
- dual of repetition code with length 6 [i]
- linear OA(327, 81, F3, 11) (dual of [81, 54, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(35, 6, F3, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(332, 87, F3, 11) (dual of [87, 55, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(332, 85, F3, 11) (dual of [85, 53, 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(332, 85, F3, 2, 11) (dual of [(85, 2), 138, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
(243−22, 243, large)-Net in Base 3 — Upper bound on s
There is no (221, 243, large)-net in base 3, because
- 20 times m-reduction [i] would yield (221, 223, large)-net in base 3, but