Best Known (172, 172+36, s)-Nets in Base 3
(172, 172+36, 1480)-Net over F3 — Constructive and digital
Digital (172, 208, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 52, 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
(172, 172+36, 5403)-Net over F3 — Digital
Digital (172, 208, 5403)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3208, 5403, F3, 36) (dual of [5403, 5195, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3208, 6616, F3, 36) (dual of [6616, 6408, 37]-code), using
- 1 times truncation [i] based on linear OA(3209, 6617, F3, 37) (dual of [6617, 6408, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [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(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(316, 56, F3, 7) (dual of [56, 40, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(3209, 6617, F3, 37) (dual of [6617, 6408, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3208, 6616, F3, 36) (dual of [6616, 6408, 37]-code), using
(172, 172+36, 1231671)-Net in Base 3 — Upper bound on s
There is no (172, 208, 1231672)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1742 694278 815879 352188 774888 069978 229182 688056 148521 643907 505936 540222 402326 525677 135491 989913 104081 > 3208 [i]