Best Known (133−32, 133, s)-Nets in Base 3
(133−32, 133, 400)-Net over F3 — Constructive and digital
Digital (101, 133, 400)-net over F3, using
- 31 times duplication [i] based on digital (100, 132, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 33, 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, 33, 100)-net over F81, using
(133−32, 133, 731)-Net over F3 — Digital
Digital (101, 133, 731)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3133, 731, F3, 32) (dual of [731, 598, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3133, 764, F3, 32) (dual of [764, 631, 33]-code), using
- construction XX applied to C1 = C([334,363]), C2 = C([340,365]), C3 = C1 + C2 = C([340,363]), and C∩ = C1 ∩ C2 = C([334,365]) [i] based on
- linear OA(3117, 728, F3, 30) (dual of [728, 611, 31]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {334,335,…,363}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3103, 728, F3, 26) (dual of [728, 625, 27]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {340,341,…,365}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {334,335,…,365}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(396, 728, F3, 24) (dual of [728, 632, 25]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {340,341,…,363}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 8, F3, 1) (dual of [8, 7, 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
- construction XX applied to C1 = C([334,363]), C2 = C([340,365]), C3 = C1 + C2 = C([340,363]), and C∩ = C1 ∩ C2 = C([334,365]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3133, 764, F3, 32) (dual of [764, 631, 33]-code), using
(133−32, 133, 31431)-Net in Base 3 — Upper bound on s
There is no (101, 133, 31432)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2865 745655 766023 625871 800971 862507 048393 981008 802965 632689 910785 > 3133 [i]