Best Known (173−36, 173, s)-Nets in Base 3
(173−36, 173, 688)-Net over F3 — Constructive and digital
Digital (137, 173, 688)-net over F3, using
- 31 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
(173−36, 173, 1723)-Net over F3 — Digital
Digital (137, 173, 1723)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3173, 1723, F3, 36) (dual of [1723, 1550, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3173, 2207, F3, 36) (dual of [2207, 2034, 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, 17, F3, 1) (dual of [17, 16, 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, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3173, 2207, F3, 36) (dual of [2207, 2034, 37]-code), using
(173−36, 173, 145449)-Net in Base 3 — Upper bound on s
There is no (137, 173, 145450)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 34832 641228 538096 779187 854885 055015 319698 567215 666999 956307 236384 140835 102921 214821 > 3173 [i]