Best Known (70−20, 70, s)-Nets in Base 3
(70−20, 70, 156)-Net over F3 — Constructive and digital
Digital (50, 70, 156)-net over F3, using
- 31 times duplication [i] based on digital (49, 69, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 23, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- trace code for nets [i] based on digital (3, 23, 52)-net over F27, using
(70−20, 70, 240)-Net over F3 — Digital
Digital (50, 70, 240)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(370, 240, F3, 20) (dual of [240, 170, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(370, 262, F3, 20) (dual of [262, 192, 21]-code), using
- construction XX applied to C1 = C([103,120]), C2 = C([106,122]), C3 = C1 + C2 = C([106,120]), and C∩ = C1 ∩ C2 = C([103,122]) [i] based on
- linear OA(360, 242, F3, 18) (dual of [242, 182, 19]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {103,104,…,120}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {106,107,…,122}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(366, 242, F3, 20) (dual of [242, 176, 21]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {103,104,…,122}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(350, 242, F3, 15) (dual of [242, 192, 16]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {106,107,…,120}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [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
- construction XX applied to C1 = C([103,120]), C2 = C([106,122]), C3 = C1 + C2 = C([106,120]), and C∩ = C1 ∩ C2 = C([103,122]) [i] based on
- discarding factors / shortening the dual code based on linear OA(370, 262, F3, 20) (dual of [262, 192, 21]-code), using
(70−20, 70, 4942)-Net in Base 3 — Upper bound on s
There is no (50, 70, 4943)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2504 811600 186576 639744 543188 192397 > 370 [i]