Best Known (151, 151+39, s)-Nets in Base 3
(151, 151+39, 688)-Net over F3 — Constructive and digital
Digital (151, 190, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (151, 192, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 172)-net over F81, using
(151, 151+39, 1970)-Net over F3 — Digital
Digital (151, 190, 1970)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3190, 1970, F3, 39) (dual of [1970, 1780, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3190, 2216, F3, 39) (dual of [2216, 2026, 40]-code), using
- construction XX applied to Ce(39) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-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(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(39) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3190, 2216, F3, 39) (dual of [2216, 2026, 40]-code), using
(151, 151+39, 220927)-Net in Base 3 — Upper bound on s
There is no (151, 190, 220928)-net in base 3, because
- 1 times m-reduction [i] would yield (151, 189, 220928)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 499445 455452 097145 849606 094335 637933 711701 611267 518420 049959 908987 768613 241151 312149 322753 > 3189 [i]