Best Known (137−27, 137, s)-Nets in Base 3
(137−27, 137, 688)-Net over F3 — Constructive and digital
Digital (110, 137, 688)-net over F3, using
- 31 times duplication [i] based on digital (109, 136, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 172)-net over F81, using
(137−27, 137, 1982)-Net over F3 — Digital
Digital (110, 137, 1982)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3137, 1982, F3, 27) (dual of [1982, 1845, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3137, 2224, F3, 27) (dual of [2224, 2087, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3127, 2188, F3, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(399, 2188, F3, 21) (dual of [2188, 2089, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3137, 2224, F3, 27) (dual of [2224, 2087, 28]-code), using
(137−27, 137, 277829)-Net in Base 3 — Upper bound on s
There is no (110, 137, 277830)-net in base 3, because
- 1 times m-reduction [i] would yield (110, 136, 277830)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 77355 720799 951635 200363 155290 831943 369088 418396 413168 361190 811989 > 3136 [i]