Best Known (73−20, 73, s)-Nets in Base 3
(73−20, 73, 192)-Net over F3 — Constructive and digital
Digital (53, 73, 192)-net over F3, using
- 31 times duplication [i] based on digital (52, 72, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 24, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- trace code for nets [i] based on digital (4, 24, 64)-net over F27, using
(73−20, 73, 285)-Net over F3 — Digital
Digital (53, 73, 285)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(373, 285, F3, 20) (dual of [285, 212, 21]-code), using
- 26 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0) [i] based on linear OA(367, 253, F3, 20) (dual of [253, 186, 21]-code), using
- construction XX applied to C1 = C([103,121]), C2 = C([105,122]), C3 = C1 + C2 = C([105,121]), and C∩ = C1 ∩ C2 = C([103,122]) [i] based on
- linear OA(361, 242, F3, 19) (dual of [242, 181, 20]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {103,104,…,121}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(361, 242, F3, 18) (dual of [242, 181, 19]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {105,106,…,122}, and designed minimum distance d ≥ |I|+1 = 19 [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(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 = {105,106,…,121}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(31, 6, F3, 1) (dual of [6, 5, 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, 5, F3, 0) (dual of [5, 5, 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([103,121]), C2 = C([105,122]), C3 = C1 + C2 = C([105,121]), and C∩ = C1 ∩ C2 = C([103,122]) [i] based on
- 26 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0) [i] based on linear OA(367, 253, F3, 20) (dual of [253, 186, 21]-code), using
(73−20, 73, 6875)-Net in Base 3 — Upper bound on s
There is no (53, 73, 6876)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 67590 908346 474118 718458 505174 239049 > 373 [i]