Best Known (162, 162+36, s)-Nets in Base 3
(162, 162+36, 896)-Net over F3 — Constructive and digital
Digital (162, 198, 896)-net over F3, using
- 32 times duplication [i] based on digital (160, 196, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 49, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 49, 224)-net over F81, using
(162, 162+36, 3903)-Net over F3 — Digital
Digital (162, 198, 3903)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3198, 3903, F3, 36) (dual of [3903, 3705, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 6585, F3, 36) (dual of [6585, 6387, 37]-code), using
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ Ce(31) [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(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(31, 4, F3, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3198, 6585, F3, 36) (dual of [6585, 6387, 37]-code), using
(162, 162+36, 668994)-Net in Base 3 — Upper bound on s
There is no (162, 198, 668995)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 29513 117105 348607 246333 934110 155498 194653 457947 693736 282666 989380 195405 078058 986975 450301 498885 > 3198 [i]