Best Known (52, 52+18, s)-Nets in Base 3
(52, 52+18, 204)-Net over F3 — Constructive and digital
Digital (52, 70, 204)-net over F3, using
- 31 times duplication [i] based on digital (51, 69, 204)-net over F3, using
- trace code for nets [i] based on digital (5, 23, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- trace code for nets [i] based on digital (5, 23, 68)-net over F27, using
(52, 52+18, 375)-Net over F3 — Digital
Digital (52, 70, 375)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(370, 375, F3, 18) (dual of [375, 305, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(370, 376, F3, 18) (dual of [376, 306, 19]-code), using
- construction XX applied to C1 = C([363,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([363,16]) [i] based on
- linear OA(364, 364, F3, 17) (dual of [364, 300, 18]-code), using the BCH-code C(I) with length 364 | 36−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(364, 364, F3, 17) (dual of [364, 300, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(370, 364, F3, 18) (dual of [364, 294, 19]-code), using the BCH-code C(I) with length 364 | 36−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(358, 364, F3, 16) (dual of [364, 306, 17]-code), using the expurgated narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 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
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code) (see above)
- construction XX applied to C1 = C([363,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([363,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(370, 376, F3, 18) (dual of [376, 306, 19]-code), using
(52, 52+18, 10649)-Net in Base 3 — Upper bound on s
There is no (52, 70, 10650)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2504 786365 975619 488108 206875 324261 > 370 [i]