Best Known (104, 137, s)-Nets in Base 3
(104, 137, 400)-Net over F3 — Constructive and digital
Digital (104, 137, 400)-net over F3, using
- 31 times duplication [i] based on digital (103, 136, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 34, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 34, 100)-net over F81, using
(104, 137, 742)-Net over F3 — Digital
Digital (104, 137, 742)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3137, 742, F3, 33) (dual of [742, 605, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3137, 757, F3, 33) (dual of [757, 620, 34]-code), using
- construction XX applied to C1 = C([724,27]), C2 = C([1,28]), C3 = C1 + C2 = C([1,27]), and C∩ = C1 ∩ C2 = C([724,28]) [i] based on
- linear OA(3127, 728, F3, 32) (dual of [728, 601, 33]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,27}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3111, 728, F3, 28) (dual of [728, 617, 29]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3130, 728, F3, 33) (dual of [728, 598, 34]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,28}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3108, 728, F3, 27) (dual of [728, 620, 28]-code), using the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(30, 3, F3, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([724,27]), C2 = C([1,28]), C3 = C1 + C2 = C([1,27]), and C∩ = C1 ∩ C2 = C([724,28]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3137, 757, F3, 33) (dual of [757, 620, 34]-code), using
(104, 137, 38624)-Net in Base 3 — Upper bound on s
There is no (104, 137, 38625)-net in base 3, because
- 1 times m-reduction [i] would yield (104, 136, 38625)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 77364 846466 519795 581528 505937 877147 113502 022943 305443 826427 604801 > 3136 [i]