Best Known (102, 102+27, s)-Nets in Base 3
(102, 102+27, 640)-Net over F3 — Constructive and digital
Digital (102, 129, 640)-net over F3, using
- 31 times duplication [i] based on digital (101, 128, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 32, 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, 32, 160)-net over F81, using
(102, 102+27, 1388)-Net over F3 — Digital
Digital (102, 129, 1388)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3129, 1388, F3, 27) (dual of [1388, 1259, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3129, 2204, F3, 27) (dual of [2204, 2075, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3128, 2203, F3, 27) (dual of [2203, 2075, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3127, 2188, F3, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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(31, 15, F3, 1) (dual of [15, 14, 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
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3128, 2203, F3, 27) (dual of [2203, 2075, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3129, 2204, F3, 27) (dual of [2204, 2075, 28]-code), using
(102, 102+27, 141301)-Net in Base 3 — Upper bound on s
There is no (102, 129, 141302)-net in base 3, because
- 1 times m-reduction [i] would yield (102, 128, 141302)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 11 790631 049212 617004 846104 073870 660777 084026 939721 564594 663989 > 3128 [i]