Best Known (131, 131+28, s)-Nets in Base 3
(131, 131+28, 701)-Net over F3 — Constructive and digital
Digital (131, 159, 701)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 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 (112, 140, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 35, 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, 35, 172)-net over F81, using
- digital (5, 19, 13)-net over F3, using
(131, 131+28, 4160)-Net over F3 — Digital
Digital (131, 159, 4160)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3159, 4160, F3, 28) (dual of [4160, 4001, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3159, 6576, F3, 28) (dual of [6576, 6417, 29]-code), using
- (u, u+v)-construction [i] based on
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(3159, 6576, F3, 28) (dual of [6576, 6417, 29]-code), using
(131, 131+28, 792783)-Net in Base 3 — Upper bound on s
There is no (131, 159, 792784)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 7282 548644 129279 007991 966894 488146 667612 597404 682649 742077 631861 107747 884513 > 3159 [i]