Best Known (108−22, 108, s)-Nets in Base 3
(108−22, 108, 640)-Net over F3 — Constructive and digital
Digital (86, 108, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 27, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
(108−22, 108, 1464)-Net over F3 — Digital
Digital (86, 108, 1464)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3108, 1464, F3, 22) (dual of [1464, 1356, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3108, 2218, F3, 22) (dual of [2218, 2110, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(378, 2187, F3, 17) (dual of [2187, 2109, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(371, 2187, F3, 16) (dual of [2187, 2116, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(38, 30, F3, 4) (dual of [30, 22, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(21) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3108, 2218, F3, 22) (dual of [2218, 2110, 23]-code), using
(108−22, 108, 118688)-Net in Base 3 — Upper bound on s
There is no (86, 108, 118689)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3381 520338 527019 881991 080620 472894 563966 570631 860603 > 3108 [i]