Best Known (145, 145+32, s)-Nets in Base 3
(145, 145+32, 701)-Net over F3 — Constructive and digital
Digital (145, 177, 701)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 21, 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 (124, 156, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 39, 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, 39, 172)-net over F81, using
- digital (5, 21, 13)-net over F3, using
(145, 145+32, 3764)-Net over F3 — Digital
Digital (145, 177, 3764)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3177, 3764, F3, 32) (dual of [3764, 3587, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3177, 6594, F3, 32) (dual of [6594, 6417, 33]-code), using
- construction XX applied to Ce(31) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(3137, 6561, F3, 26) (dual of [6561, 6424, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(36, 31, F3, 3) (dual of [31, 25, 4]-code or 31-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(31) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3177, 6594, F3, 32) (dual of [6594, 6417, 33]-code), using
(145, 145+32, 645137)-Net in Base 3 — Upper bound on s
There is no (145, 177, 645138)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2 821429 859642 862056 014683 618008 680030 983482 673452 796980 184080 044496 610127 166526 957217 > 3177 [i]