Best Known (165, 203, s)-Nets in Base 3
(165, 203, 700)-Net over F3 — Constructive and digital
Digital (165, 203, 700)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 23, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (142, 180, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 45, 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, 45, 172)-net over F81, using
- digital (4, 23, 12)-net over F3, using
(165, 203, 3361)-Net over F3 — Digital
Digital (165, 203, 3361)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3203, 3361, F3, 38) (dual of [3361, 3158, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3203, 6572, F3, 38) (dual of [6572, 6369, 39]-code), using
- construction XX applied to Ce(37) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- 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(3185, 6561, F3, 35) (dual of [6561, 6376, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(37) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3203, 6572, F3, 38) (dual of [6572, 6369, 39]-code), using
(165, 203, 496401)-Net in Base 3 — Upper bound on s
There is no (165, 203, 496402)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 7 171660 746875 557992 061912 942562 789186 907388 448792 062306 332067 751565 641126 800412 350594 700242 003569 > 3203 [i]