Best Known (151−31, 151, s)-Nets in Base 3
(151−31, 151, 640)-Net over F3 — Constructive and digital
Digital (120, 151, 640)-net over F3, using
- t-expansion [i] based on digital (119, 151, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (119, 152, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 38, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 38, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (119, 152, 640)-net over F3, using
(151−31, 151, 1687)-Net over F3 — Digital
Digital (120, 151, 1687)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3151, 1687, F3, 31) (dual of [1687, 1536, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, 2224, F3, 31) (dual of [2224, 2073, 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, 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([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3151, 2224, F3, 31) (dual of [2224, 2073, 32]-code), using
(151−31, 151, 189633)-Net in Base 3 — Upper bound on s
There is no (120, 151, 189634)-net in base 3, because
- 1 times m-reduction [i] would yield (120, 150, 189634)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 369991 280047 864639 863518 305484 117539 811575 128765 444454 126057 040892 642201 > 3150 [i]