Best Known (150−30, 150, s)-Nets in Base 3
(150−30, 150, 688)-Net over F3 — Constructive and digital
Digital (120, 150, 688)-net over F3, using
- 32 times duplication [i] based on digital (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 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, 37, 172)-net over F81, using
(150−30, 150, 1928)-Net over F3 — Digital
Digital (120, 150, 1928)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3150, 1928, F3, 30) (dual of [1928, 1778, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, 2225, F3, 30) (dual of [2225, 2075, 31]-code), using
- 1 times truncation [i] based on linear OA(3151, 2226, F3, 31) (dual of [2226, 2075, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3113, 2188, F3, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- 1 times truncation [i] based on linear OA(3151, 2226, F3, 31) (dual of [2226, 2075, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, 2225, F3, 30) (dual of [2225, 2075, 31]-code), using
(150−30, 150, 189633)-Net in Base 3 — Upper bound on s
There is no (120, 150, 189634)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 369991 280047 864639 863518 305484 117539 811575 128765 444454 126057 040892 642201 > 3150 [i]