Best Known (215−37, 215, s)-Nets in Base 3
(215−37, 215, 1480)-Net over F3 — Constructive and digital
Digital (178, 215, 1480)-net over F3, using
- 1 times m-reduction [i] based on digital (178, 216, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 54, 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, 54, 370)-net over F81, using
(215−37, 215, 5714)-Net over F3 — Digital
Digital (178, 215, 5714)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3215, 5714, F3, 37) (dual of [5714, 5499, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, 6631, F3, 37) (dual of [6631, 6416, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3214, 6630, F3, 37) (dual of [6630, 6416, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(27) [i] based on
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(36) ⊂ Ce(27) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3214, 6630, F3, 37) (dual of [6630, 6416, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, 6631, F3, 37) (dual of [6631, 6416, 38]-code), using
(215−37, 215, 1776386)-Net in Base 3 — Upper bound on s
There is no (178, 215, 1776387)-net in base 3, because
- 1 times m-reduction [i] would yield (178, 214, 1776387)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 270435 044967 065208 996638 674837 585023 360470 042666 942568 996263 555557 015192 126651 497099 163696 981964 762501 > 3214 [i]