Best Known (141−24, 141, s)-Nets in Base 3
(141−24, 141, 701)-Net over F3 — Constructive and digital
Digital (117, 141, 701)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 17, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- digital (100, 124, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 31, 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, 31, 172)-net over F81, using
- digital (5, 17, 13)-net over F3, using
(141−24, 141, 4900)-Net over F3 — Digital
Digital (117, 141, 4900)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3141, 4900, F3, 24) (dual of [4900, 4759, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3141, 6606, F3, 24) (dual of [6606, 6465, 25]-code), using
- strength reduction [i] based on linear OA(3141, 6606, F3, 25) (dual of [6606, 6465, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,12]) ⊂ C([0,9]) [i] based on
- strength reduction [i] based on linear OA(3141, 6606, F3, 25) (dual of [6606, 6465, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3141, 6606, F3, 24) (dual of [6606, 6465, 25]-code), using
(141−24, 141, 1067828)-Net in Base 3 — Upper bound on s
There is no (117, 141, 1067829)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 18 797480 524982 623935 383540 546581 460966 520965 498671 176910 549972 667313 > 3141 [i]