Best Known (189, 189+52, s)-Nets in Base 3
(189, 189+52, 688)-Net over F3 — Constructive and digital
Digital (189, 241, 688)-net over F3, using
- 31 times duplication [i] based on digital (188, 240, 688)-net over F3, using
- t-expansion [i] based on digital (187, 240, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 60, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 60, 172)-net over F81, using
- t-expansion [i] based on digital (187, 240, 688)-net over F3, using
(189, 189+52, 1854)-Net over F3 — Digital
Digital (189, 241, 1854)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3241, 1854, F3, 52) (dual of [1854, 1613, 53]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, 2197, F3, 52) (dual of [2197, 1956, 53]-code), using
- construction XX applied to Ce(51) ⊂ Ce(49) ⊂ Ce(48) [i] based on
- linear OA(3239, 2187, F3, 52) (dual of [2187, 1948, 53]-code), using an extension Ce(51) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,51], and designed minimum distance d ≥ |I|+1 = 52 [i]
- linear OA(3232, 2187, F3, 50) (dual of [2187, 1955, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(3225, 2187, F3, 49) (dual of [2187, 1962, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(51) ⊂ Ce(49) ⊂ Ce(48) [i] based on
- discarding factors / shortening the dual code based on linear OA(3241, 2197, F3, 52) (dual of [2197, 1956, 53]-code), using
(189, 189+52, 139550)-Net in Base 3 — Upper bound on s
There is no (189, 241, 139551)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9 689036 729473 709889 402999 890299 282386 513426 714052 256999 095266 913627 008036 085728 785450 849723 961851 685128 446543 589901 > 3241 [i]