Best Known (58, 58+15, s)-Nets in Base 3
(58, 58+15, 464)-Net over F3 — Constructive and digital
Digital (58, 73, 464)-net over F3, using
- 31 times duplication [i] based on digital (57, 72, 464)-net over F3, using
- t-expansion [i] based on digital (56, 72, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 18, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 18, 116)-net over F81, using
- t-expansion [i] based on digital (56, 72, 464)-net over F3, using
(58, 58+15, 1232)-Net over F3 — Digital
Digital (58, 73, 1232)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(373, 1232, F3, 15) (dual of [1232, 1159, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 2205, F3, 15) (dual of [2205, 2132, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(372, 2204, F3, 15) (dual of [2204, 2132, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(371, 2188, F3, 15) (dual of [2188, 2117, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(357, 2188, F3, 13) (dual of [2188, 2131, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(315, 16, F3, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
- dual of repetition code with length 16 [i]
- linear OA(31, 16, F3, 1) (dual of [16, 15, 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 X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(372, 2204, F3, 15) (dual of [2204, 2132, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 2205, F3, 15) (dual of [2205, 2132, 16]-code), using
(58, 58+15, 136584)-Net in Base 3 — Upper bound on s
There is no (58, 73, 136585)-net in base 3, because
- 1 times m-reduction [i] would yield (58, 72, 136585)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 22529 167844 931063 017505 600265 338795 > 372 [i]