Best Known (224−23, 224, s)-Nets in Base 3
(224−23, 224, 434824)-Net over F3 — Constructive and digital
Digital (201, 224, 434824)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 13, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-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 (2, 13, 8)-net over F3, using
(224−23, 224, 1405573)-Net over F3 — Digital
Digital (201, 224, 1405573)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3224, 1405573, F3, 3, 23) (dual of [(1405573, 3), 4216495, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3224, 1594346, F3, 3, 23) (dual of [(1594346, 3), 4782814, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3224, 4783038, F3, 23) (dual of [4783038, 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(313, 69, F3, 5) (dual of [69, 56, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(313, 80, F3, 5) (dual of [80, 67, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(313, 80, F3, 5) (dual of [80, 67, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(3224, 4783038, F3, 23) (dual of [4783038, 4782814, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(3224, 1594346, F3, 3, 23) (dual of [(1594346, 3), 4782814, 24]-NRT-code), using
(224−23, 224, large)-Net in Base 3 — Upper bound on s
There is no (201, 224, large)-net in base 3, because
- 21 times m-reduction [i] would yield (201, 203, large)-net in base 3, but