Best Known (171−32, 171, s)-Nets in Base 3
(171−32, 171, 688)-Net over F3 — Constructive and digital
Digital (139, 171, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (139, 176, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 44, 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, 44, 172)-net over F81, using
(171−32, 171, 3286)-Net over F3 — Digital
Digital (139, 171, 3286)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3171, 3286, F3, 2, 32) (dual of [(3286, 2), 6401, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3171, 6572, F3, 32) (dual of [6572, 6401, 33]-code), using
- construction XX applied to Ce(31) ⊂ Ce(30) ⊂ Ce(28) [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(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(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 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
- 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(30) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(3171, 6572, F3, 32) (dual of [6572, 6401, 33]-code), using
(171−32, 171, 427293)-Net in Base 3 — Upper bound on s
There is no (139, 171, 427294)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3870 268582 043245 338110 059501 686005 657279 732069 670626 083489 555690 160069 523958 422945 > 3171 [i]