Best Known (52, 52+20, s)-Nets in Base 3
(52, 52+20, 192)-Net over F3 — Constructive and digital
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
(52, 52+20, 273)-Net over F3 — Digital
Digital (52, 72, 273)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(372, 273, F3, 20) (dual of [273, 201, 21]-code), using
- 19 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0) [i] based on linear OA(366, 248, F3, 20) (dual of [248, 182, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(366, 243, F3, 20) (dual of [243, 177, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(361, 243, F3, 19) (dual of [243, 182, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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 X applied to Ce(19) ⊂ Ce(18) [i] based on
- 19 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0) [i] based on linear OA(366, 248, F3, 20) (dual of [248, 182, 21]-code), using
(52, 52+20, 6159)-Net in Base 3 — Upper bound on s
There is no (52, 72, 6160)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 22544 068283 891676 285559 766486 061921 > 372 [i]