Best Known (78−20, 78, s)-Nets in Base 3
(78−20, 78, 228)-Net over F3 — Constructive and digital
Digital (58, 78, 228)-net over F3, using
- trace code for nets [i] based on digital (6, 26, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
(78−20, 78, 385)-Net over F3 — Digital
Digital (58, 78, 385)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(378, 385, F3, 20) (dual of [385, 307, 21]-code), using
- 7 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0) [i] 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
- 7 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0) [i] based on linear OA(377, 377, F3, 20) (dual of [377, 300, 21]-code), using
(78−20, 78, 11916)-Net in Base 3 — Upper bound on s
There is no (58, 78, 11917)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 16 431071 782225 647132 422777 624547 770817 > 378 [i]