Best Known (171, 211, s)-Nets in Base 3
(171, 211, 698)-Net over F3 — Constructive and digital
Digital (171, 211, 698)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 23, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (148, 188, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 47, 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, 47, 172)-net over F81, using
- digital (3, 23, 10)-net over F3, using
(171, 211, 3286)-Net over F3 — Digital
Digital (171, 211, 3286)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3211, 3286, F3, 2, 40) (dual of [(3286, 2), 6361, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3211, 6572, F3, 40) (dual of [6572, 6361, 41]-code), using
- construction XX applied to Ce(39) ⊂ Ce(37) ⊂ Ce(36) [i] based on
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(39) ⊂ Ce(37) ⊂ Ce(36) [i] based on
- OOA 2-folding [i] based on linear OA(3211, 6572, F3, 40) (dual of [6572, 6361, 41]-code), using
(171, 211, 448628)-Net in Base 3 — Upper bound on s
There is no (171, 211, 448629)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 47054 340531 500507 104069 243491 017562 519040 525428 526791 124037 868845 166963 783918 490656 350690 559331 973521 > 3211 [i]