Best Known (174−36, 174, s)-Nets in Base 3
(174−36, 174, 688)-Net over F3 — Constructive and digital
Digital (138, 174, 688)-net over F3, using
- 32 times duplication [i] based on digital (136, 172, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 43, 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, 43, 172)-net over F81, using
(174−36, 174, 1780)-Net over F3 — Digital
Digital (138, 174, 1780)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3174, 1780, F3, 36) (dual of [1780, 1606, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3174, 2209, F3, 36) (dual of [2209, 2035, 37]-code), using
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3148, 2187, F3, 32) (dual of [2187, 2039, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(31, 18, F3, 1) (dual of [18, 17, 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(3174, 2209, F3, 36) (dual of [2209, 2035, 37]-code), using
(174−36, 174, 154604)-Net in Base 3 — Upper bound on s
There is no (138, 174, 154605)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 104497 835522 235830 987910 671103 296792 212476 007606 578198 812427 457506 250303 501858 437537 > 3174 [i]