Best Known (177, 177+44, s)-Nets in Base 3
(177, 177+44, 688)-Net over F3 — Constructive and digital
Digital (177, 221, 688)-net over F3, using
- t-expansion [i] based on digital (175, 221, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (175, 224, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 56, 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, 56, 172)-net over F81, using
- 3 times m-reduction [i] based on digital (175, 224, 688)-net over F3, using
(177, 177+44, 2432)-Net over F3 — Digital
Digital (177, 221, 2432)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3221, 2432, F3, 44) (dual of [2432, 2211, 45]-code), using
- 221 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 26 times 0, 1, 32 times 0, 1, 40 times 0, 1, 46 times 0) [i] based on linear OA(3204, 2194, F3, 44) (dual of [2194, 1990, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- 221 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 26 times 0, 1, 32 times 0, 1, 40 times 0, 1, 46 times 0) [i] based on linear OA(3204, 2194, F3, 44) (dual of [2194, 1990, 45]-code), using
(177, 177+44, 280989)-Net in Base 3 — Upper bound on s
There is no (177, 221, 280990)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2778 582769 685705 302914 499868 822989 994502 021340 823434 054893 455680 410522 577910 372593 487946 526718 104057 778949 > 3221 [i]