Best Known (180, 227, s)-Nets in Base 3
(180, 227, 688)-Net over F3 — Constructive and digital
Digital (180, 227, 688)-net over F3, using
- t-expansion [i] based on digital (178, 227, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (178, 228, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 57, 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, 57, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (178, 228, 688)-net over F3, using
(180, 227, 2153)-Net over F3 — Digital
Digital (180, 227, 2153)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 2153, F3, 47) (dual of [2153, 1926, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3227, 2219, F3, 47) (dual of [2219, 1992, 48]-code), using
- construction XX applied to Ce(46) ⊂ Ce(42) ⊂ Ce(40) [i] based on
- linear OA(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [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(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(36, 29, F3, 3) (dual of [29, 23, 4]-code or 29-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
- linear OA(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(46) ⊂ Ce(42) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3227, 2219, F3, 47) (dual of [2219, 1992, 48]-code), using
(180, 227, 229942)-Net in Base 3 — Upper bound on s
There is no (180, 227, 229943)-net in base 3, because
- 1 times m-reduction [i] would yield (180, 226, 229943)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 675207 459932 214652 257421 949154 568315 605444 233689 953317 485995 628466 514790 370235 631330 964705 722745 778659 542323 > 3226 [i]