Best Known (119, 119+38, s)-Nets in Base 3
(119, 119+38, 400)-Net over F3 — Constructive and digital
Digital (119, 157, 400)-net over F3, using
- 31 times duplication [i] based on digital (118, 156, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 39, 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, 39, 100)-net over F81, using
(119, 119+38, 799)-Net over F3 — Digital
Digital (119, 157, 799)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3157, 799, F3, 38) (dual of [799, 642, 39]-code), using
- 50 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 12 times 0, 1, 15 times 0) [i] based on linear OA(3149, 741, F3, 38) (dual of [741, 592, 39]-code), using
- construction XX applied to C1 = C([328,364]), C2 = C([330,365]), C3 = C1 + C2 = C([330,364]), and C∩ = C1 ∩ C2 = C([328,365]) [i] based on
- linear OA(3142, 728, F3, 37) (dual of [728, 586, 38]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {328,329,…,364}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3142, 728, F3, 36) (dual of [728, 586, 37]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {330,331,…,365}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3148, 728, F3, 38) (dual of [728, 580, 39]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {328,329,…,365}, and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(3136, 728, F3, 35) (dual of [728, 592, 36]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {330,331,…,364}, and designed minimum distance d ≥ |I|+1 = 36 [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([328,364]), C2 = C([330,365]), C3 = C1 + C2 = C([330,364]), and C∩ = C1 ∩ C2 = C([328,365]) [i] based on
- 50 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 12 times 0, 1, 15 times 0) [i] based on linear OA(3149, 741, F3, 38) (dual of [741, 592, 39]-code), using
(119, 119+38, 34712)-Net in Base 3 — Upper bound on s
There is no (119, 157, 34713)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 809 508812 727979 425130 457020 171818 975162 466619 592330 151943 945066 289572 766459 > 3157 [i]