Best Known (152, 152+35, s)-Nets in Base 3
(152, 152+35, 696)-Net over F3 — Constructive and digital
Digital (152, 187, 696)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 19, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
- digital (2, 19, 8)-net over F3, using
(152, 152+35, 3286)-Net over F3 — Digital
Digital (152, 187, 3286)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3187, 3286, F3, 2, 35) (dual of [(3286, 2), 6385, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3187, 6572, F3, 35) (dual of [6572, 6385, 36]-code), using
- construction XX applied to Ce(34) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- 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(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(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(34) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- OOA 2-folding [i] based on linear OA(3187, 6572, F3, 35) (dual of [6572, 6385, 36]-code), using
(152, 152+35, 595896)-Net in Base 3 — Upper bound on s
There is no (152, 187, 595897)-net in base 3, because
- 1 times m-reduction [i] would yield (152, 186, 595897)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 55533 650110 001749 516045 457151 470503 394255 871196 732341 129200 054068 506457 614515 552437 140531 > 3186 [i]