Best Known (53, 53+18, s)-Nets in Base 3
(53, 53+18, 204)-Net over F3 — Constructive and digital
Digital (53, 71, 204)-net over F3, using
- 1 times m-reduction [i] based on digital (53, 72, 204)-net over F3, using
- trace code for nets [i] based on digital (5, 24, 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, 24, 68)-net over F27, using
(53, 53+18, 383)-Net over F3 — Digital
Digital (53, 71, 383)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(371, 383, F3, 18) (dual of [383, 312, 19]-code), using
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] 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
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] based on linear OA(370, 376, F3, 18) (dual of [376, 306, 19]-code), using
(53, 53+18, 12032)-Net in Base 3 — Upper bound on s
There is no (53, 71, 12033)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 7509 852735 017487 621304 413306 690275 > 371 [i]