Best Known (130, 130+27, s)-Nets in Base 3
(130, 130+27, 704)-Net over F3 — Constructive and digital
Digital (130, 157, 704)-net over F3, using
- 31 times duplication [i] based on digital (129, 156, 704)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 16)-net over F3, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 7 and N(F) ≥ 16, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- digital (109, 136, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 172)-net over F81, using
- digital (7, 20, 16)-net over F3, using
- (u, u+v)-construction [i] based on
(130, 130+27, 4805)-Net over F3 — Digital
Digital (130, 157, 4805)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3157, 4805, F3, 27) (dual of [4805, 4648, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 6606, F3, 27) (dual of [6606, 6449, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3145, 6562, F3, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3113, 6562, F3, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(312, 44, F3, 5) (dual of [44, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3157, 6606, F3, 27) (dual of [6606, 6449, 28]-code), using
(130, 130+27, 1506009)-Net in Base 3 — Upper bound on s
There is no (130, 157, 1506010)-net in base 3, because
- 1 times m-reduction [i] would yield (130, 156, 1506010)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 269 723390 791506 168465 075223 312939 216441 728155 291825 349367 154741 663046 565821 > 3156 [i]