Best Known (57, 57+20, s)-Nets in Base 3
(57, 57+20, 204)-Net over F3 — Constructive and digital
Digital (57, 77, 204)-net over F3, using
- 1 times m-reduction [i] based on digital (57, 78, 204)-net over F3, using
- trace code for nets [i] based on digital (5, 26, 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, 26, 68)-net over F27, using
(57, 57+20, 375)-Net over F3 — Digital
Digital (57, 77, 375)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(377, 375, F3, 20) (dual of [375, 298, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(377, 377, F3, 20) (dual of [377, 300, 21]-code), using
- construction XX applied to C1 = C([164,182]), C2 = C([166,183]), C3 = C1 + C2 = C([166,182]), and C∩ = C1 ∩ C2 = C([164,183]) [i] based on
- linear OA(370, 364, F3, 19) (dual of [364, 294, 20]-code), using the BCH-code C(I) with length 364 | 36−1, defining interval I = {164,165,…,182}, and designed minimum distance d ≥ |I|+1 = 20 [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 = {166,167,…,183}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(376, 364, F3, 20) (dual of [364, 288, 21]-code), using the BCH-code C(I) with length 364 | 36−1, defining interval I = {164,165,…,183}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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 = {166,167,…,182}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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(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
- construction XX applied to C1 = C([164,182]), C2 = C([166,183]), C3 = C1 + C2 = C([166,182]), and C∩ = C1 ∩ C2 = C([164,183]) [i] based on
- discarding factors / shortening the dual code based on linear OA(377, 377, F3, 20) (dual of [377, 300, 21]-code), using
(57, 57+20, 10675)-Net in Base 3 — Upper bound on s
There is no (57, 77, 10676)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5 476258 878289 420319 138631 760417 170649 > 377 [i]