Best Known (224−38, 224, s)-Nets in Base 3
(224−38, 224, 1480)-Net over F3 — Constructive and digital
Digital (186, 224, 1480)-net over F3, using
- t-expansion [i] based on digital (184, 224, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 56, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 56, 370)-net over F81, using
(224−38, 224, 6410)-Net over F3 — Digital
Digital (186, 224, 6410)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3224, 6410, F3, 38) (dual of [6410, 6186, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 6632, F3, 38) (dual of [6632, 6408, 39]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3222, 6630, F3, 38) (dual of [6630, 6408, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(28) [i] based on
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(321, 69, F3, 8) (dual of [69, 48, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- construction X applied to Ce(37) ⊂ Ce(28) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3222, 6630, F3, 38) (dual of [6630, 6408, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 6632, F3, 38) (dual of [6632, 6408, 39]-code), using
(224−38, 224, 1671818)-Net in Base 3 — Upper bound on s
There is no (186, 224, 1671819)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 75017 481823 192557 989420 786414 389178 509599 020488 299421 910376 752474 416187 836183 189737 917450 141548 431277 751595 > 3224 [i]