Best Known (223−23, 223, s)-Nets in Base 3
(223−23, 223, 434823)-Net over F3 — Constructive and digital
Digital (200, 223, 434823)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (188, 211, 434816)-net over F3, using
- net defined by OOA [i] based on linear OOA(3211, 434816, F3, 23, 23) (dual of [(434816, 23), 10000557, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3211, 4782977, F3, 23) (dual of [4782977, 4782766, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, 4782983, F3, 23) (dual of [4782983, 4782772, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 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(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3211, 4782983, F3, 23) (dual of [4782983, 4782772, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3211, 4782977, F3, 23) (dual of [4782977, 4782766, 24]-code), using
- net defined by OOA [i] based on linear OOA(3211, 434816, F3, 23, 23) (dual of [(434816, 23), 10000557, 24]-NRT-code), using
- digital (1, 12, 7)-net over F3, using
(223−23, 223, 1326605)-Net over F3 — Digital
Digital (200, 223, 1326605)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3223, 1326605, F3, 3, 23) (dual of [(1326605, 3), 3979592, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3223, 1594345, F3, 3, 23) (dual of [(1594345, 3), 4782812, 24]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3222, 1594345, F3, 3, 23) (dual of [(1594345, 3), 4782813, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3222, 4783035, F3, 23) (dual of [4783035, 4782813, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 4783036, F3, 23) (dual of [4783036, 4782814, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3155, 4782969, F3, 17) (dual of [4782969, 4782814, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(311, 67, F3, 5) (dual of [67, 56, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 4783036, F3, 23) (dual of [4783036, 4782814, 24]-code), using
- OOA 3-folding [i] based on linear OA(3222, 4783035, F3, 23) (dual of [4783035, 4782813, 24]-code), using
- 31 times duplication [i] based on linear OOA(3222, 1594345, F3, 3, 23) (dual of [(1594345, 3), 4782813, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3223, 1594345, F3, 3, 23) (dual of [(1594345, 3), 4782812, 24]-NRT-code), using
(223−23, 223, large)-Net in Base 3 — Upper bound on s
There is no (200, 223, large)-net in base 3, because
- 21 times m-reduction [i] would yield (200, 202, large)-net in base 3, but