Best Known (216, 216+22, s)-Nets in Base 3
(216, 216+22, 762640)-Net over F3 — Constructive and digital
Digital (216, 238, 762640)-net over F3, using
- 31 times duplication [i] based on digital (215, 237, 762640)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (15, 26, 40)-net over F3, using
- trace code for nets [i] based on digital (2, 13, 20)-net over F9, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 2 and N(F) ≥ 20, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- trace code for nets [i] based on digital (2, 13, 20)-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 (15, 26, 40)-net over F3, using
- (u, u+v)-construction [i] based on
(216, 216+22, 3346042)-Net over F3 — Digital
Digital (216, 238, 3346042)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3238, 3346042, F3, 2, 22) (dual of [(3346042, 2), 6691846, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3238, 4194342, F3, 2, 22) (dual of [(4194342, 2), 8388446, 23]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(327, 41, F3, 2, 11) (dual of [(41, 2), 55, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(327, 82, F3, 11) (dual of [82, 55, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(327, 83, F3, 11) (dual of [83, 56, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(325, 81, F3, 10) (dual of [81, 56, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 2, F3, 0) (dual of [2, 2, 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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(327, 83, F3, 11) (dual of [83, 56, 12]-code), using
- OOA 2-folding [i] based on linear OA(327, 82, F3, 11) (dual of [82, 55, 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(327, 41, F3, 2, 11) (dual of [(41, 2), 55, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(3238, 4194342, F3, 2, 22) (dual of [(4194342, 2), 8388446, 23]-NRT-code), using
(216, 216+22, large)-Net in Base 3 — Upper bound on s
There is no (216, 238, large)-net in base 3, because
- 20 times m-reduction [i] would yield (216, 218, large)-net in base 3, but