Best Known (244−38, 244, s)-Nets in Base 3
(244−38, 244, 1499)-Net over F3 — Constructive and digital
Digital (206, 244, 1499)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (9, 28, 19)-net over F3, using
- net from sequence [i] based on digital (9, 18)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 9 and N(F) ≥ 19, using
- net from sequence [i] based on digital (9, 18)-sequence over F3, using
- digital (178, 216, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 54, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 54, 370)-net over F81, using
- digital (9, 28, 19)-net over F3, using
(244−38, 244, 11830)-Net over F3 — Digital
Digital (206, 244, 11830)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3244, 11830, F3, 38) (dual of [11830, 11586, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3244, 19729, F3, 38) (dual of [19729, 19485, 39]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- linear OA(3235, 19684, F3, 39) (dual of [19684, 19449, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3199, 19684, F3, 33) (dual of [19684, 19485, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(39, 45, F3, 4) (dual of [45, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3244, 19729, F3, 38) (dual of [19729, 19485, 39]-code), using
(244−38, 244, 5314046)-Net in Base 3 — Upper bound on s
There is no (206, 244, 5314047)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 261 569110 638738 125651 292133 928258 711519 385256 687534 191656 263131 710048 882439 063151 545461 446490 310782 299583 190880 221147 > 3244 [i]