Best Known (234−49, 234, s)-Nets in Base 3
(234−49, 234, 688)-Net over F3 — Constructive and digital
Digital (185, 234, 688)-net over F3, using
- t-expansion [i] based on digital (184, 234, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (184, 236, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 59, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 59, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (184, 236, 688)-net over F3, using
(234−49, 234, 2086)-Net over F3 — Digital
Digital (185, 234, 2086)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3234, 2086, F3, 49) (dual of [2086, 1852, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(3234, 2218, F3, 49) (dual of [2218, 1984, 50]-code), using
- construction XX applied to Ce(48) ⊂ Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3225, 2187, F3, 49) (dual of [2187, 1962, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(38, 30, F3, 4) (dual of [30, 22, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(48) ⊂ Ce(43) ⊂ Ce(42) [i] based on
- discarding factors / shortening the dual code based on linear OA(3234, 2218, F3, 49) (dual of [2218, 1984, 50]-code), using
(234−49, 234, 210056)-Net in Base 3 — Upper bound on s
There is no (185, 234, 210057)-net in base 3, because
- 1 times m-reduction [i] would yield (185, 233, 210057)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1476 631023 196683 105142 915316 932079 579564 153069 071812 946554 482326 359778 626125 255246 087977 108455 974274 142747 962849 > 3233 [i]