Best Known (193−39, 193, s)-Nets in Base 3
(193−39, 193, 688)-Net over F3 — Constructive and digital
Digital (154, 193, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (154, 196, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 49, 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, 49, 172)-net over F81, using
(193−39, 193, 2157)-Net over F3 — Digital
Digital (154, 193, 2157)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3193, 2157, F3, 39) (dual of [2157, 1964, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3193, 2224, F3, 39) (dual of [2224, 2031, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- linear OA(3183, 2188, F3, 39) (dual of [2188, 2005, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3155, 2188, F3, 33) (dual of [2188, 2033, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3193, 2224, F3, 39) (dual of [2224, 2031, 40]-code), using
(193−39, 193, 262779)-Net in Base 3 — Upper bound on s
There is no (154, 193, 262780)-net in base 3, because
- 1 times m-reduction [i] would yield (154, 192, 262780)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 40 486481 692703 443163 302548 330262 233479 047116 884425 139216 900889 463564 190527 743775 536811 899537 > 3192 [i]