Best Known (178, 178+36, s)-Nets in Base 3
(178, 178+36, 1480)-Net over F3 — Constructive and digital
Digital (178, 214, 1480)-net over F3, using
- 2 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
(178, 178+36, 6566)-Net over F3 — Digital
Digital (178, 214, 6566)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3214, 6566, F3, 36) (dual of [6566, 6352, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3214, 6631, F3, 36) (dual of [6631, 6417, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,13]) [i] based on
- linear OA(3193, 6562, F3, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3145, 6562, F3, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [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 C([0,18]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3214, 6631, F3, 36) (dual of [6631, 6417, 37]-code), using
(178, 178+36, 1776386)-Net in Base 3 — Upper bound on s
There is no (178, 214, 1776387)-net in base 3, because
- 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]