Best Known (31, 31+12, s)-Nets in Base 3
(31, 31+12, 144)-Net over F3 — Constructive and digital
Digital (31, 43, 144)-net over F3, using
- 31 times duplication [i] based on digital (30, 42, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 14, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- trace code for nets [i] based on digital (2, 14, 48)-net over F27, using
(31, 31+12, 220)-Net over F3 — Digital
Digital (31, 43, 220)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(343, 220, F3, 12) (dual of [220, 177, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 248, F3, 12) (dual of [248, 205, 13]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(341, 244, F3, 13) (dual of [244, 203, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(331, 244, F3, 9) (dual of [244, 213, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- extended Reed–Solomon code RSe(2,3) [i]
- Hamming code H(2,3) [i]
- Simplex code S(2,3) [i]
- the Tetracode [i]
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(343, 248, F3, 12) (dual of [248, 205, 13]-code), using
(31, 31+12, 3926)-Net in Base 3 — Upper bound on s
There is no (31, 43, 3927)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 328 743916 827059 177445 > 343 [i]