Best Known (145, 145+34, s)-Nets in Base 3
(145, 145+34, 688)-Net over F3 — Constructive and digital
Digital (145, 179, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (145, 184, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 46, 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, 46, 172)-net over F81, using
(145, 145+34, 3260)-Net over F3 — Digital
Digital (145, 179, 3260)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3179, 3260, F3, 2, 34) (dual of [(3260, 2), 6341, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3179, 3286, F3, 2, 34) (dual of [(3286, 2), 6393, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3179, 6572, F3, 34) (dual of [6572, 6393, 35]-code), using
- construction XX applied to Ce(33) ⊂ Ce(31) ⊂ Ce(30) [i] based on
- linear OA(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 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(3161, 6561, F3, 31) (dual of [6561, 6400, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(33) ⊂ Ce(31) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(3179, 6572, F3, 34) (dual of [6572, 6393, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(3179, 3286, F3, 2, 34) (dual of [(3286, 2), 6393, 35]-NRT-code), using
(145, 145+34, 379055)-Net in Base 3 — Upper bound on s
There is no (145, 179, 379056)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 25 393359 209213 276865 861540 960509 526971 739144 755188 967687 872555 248244 320785 913325 452129 > 3179 [i]