Best Known (217−46, 217, s)-Nets in Base 3
(217−46, 217, 688)-Net over F3 — Constructive and digital
Digital (171, 217, 688)-net over F3, using
- 31 times duplication [i] based on digital (170, 216, 688)-net over F3, using
- t-expansion [i] based on digital (169, 216, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
- t-expansion [i] based on digital (169, 216, 688)-net over F3, using
(217−46, 217, 1857)-Net over F3 — Digital
Digital (171, 217, 1857)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3217, 1857, F3, 46) (dual of [1857, 1640, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 2208, F3, 46) (dual of [2208, 1991, 47]-code), using
- construction XX applied to Ce(45) ⊂ Ce(42) ⊂ Ce(40) [i] based on
- linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(34, 19, F3, 2) (dual of [19, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(45) ⊂ Ce(42) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3217, 2208, F3, 46) (dual of [2208, 1991, 47]-code), using
(217−46, 217, 149587)-Net in Base 3 — Upper bound on s
There is no (171, 217, 149588)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 34 302287 891978 711620 341613 120848 958998 504252 833421 693416 682430 666241 238624 139184 871529 589680 526910 560049 > 3217 [i]