Best Known (150−31, 150, s)-Nets in Base 3
(150−31, 150, 640)-Net over F3 — Constructive and digital
Digital (119, 150, 640)-net over F3, using
- 2 times m-reduction [i] based on digital (119, 152, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 38, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 38, 160)-net over F81, using
(150−31, 150, 1623)-Net over F3 — Digital
Digital (119, 150, 1623)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3150, 1623, F3, 31) (dual of [1623, 1473, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, 2218, F3, 31) (dual of [2218, 2068, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3120, 2187, F3, 26) (dual of [2187, 2067, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3113, 2187, F3, 25) (dual of [2187, 2074, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(38, 30, F3, 4) (dual of [30, 22, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), 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(30) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3150, 2218, F3, 31) (dual of [2218, 2068, 32]-code), using
(150−31, 150, 176240)-Net in Base 3 — Upper bound on s
There is no (119, 150, 176241)-net in base 3, because
- 1 times m-reduction [i] would yield (119, 149, 176241)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 123336 174520 977309 864468 487017 810214 071639 450965 316101 412183 426531 663915 > 3149 [i]