Best Known (136, 167, s)-Nets in Base 3
(136, 167, 692)-Net over F3 — Constructive and digital
Digital (136, 167, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 15, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (121, 152, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 38, 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, 38, 172)-net over F81, using
- digital (0, 15, 4)-net over F3, using
(136, 167, 3292)-Net over F3 — Digital
Digital (136, 167, 3292)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3167, 3292, F3, 2, 31) (dual of [(3292, 2), 6417, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3167, 6584, F3, 31) (dual of [6584, 6417, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3145, 6562, F3, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(36, 22, F3, 3) (dual of [22, 16, 4]-code or 22-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- OOA 2-folding [i] based on linear OA(3167, 6584, F3, 31) (dual of [6584, 6417, 32]-code), using
(136, 167, 612164)-Net in Base 3 — Upper bound on s
There is no (136, 167, 612165)-net in base 3, because
- 1 times m-reduction [i] would yield (136, 166, 612165)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 15 926929 481532 286204 739729 487464 539737 672754 288955 047724 605673 453905 235497 813371 > 3166 [i]