Best Known (150−32, 150, s)-Nets in Base 3
(150−32, 150, 640)-Net over F3 — Constructive and digital
Digital (118, 150, 640)-net over F3, using
- 32 times duplication [i] based on digital (116, 148, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 37, 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, 37, 160)-net over F81, using
(150−32, 150, 1383)-Net over F3 — Digital
Digital (118, 150, 1383)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3150, 1383, F3, 32) (dual of [1383, 1233, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, 2197, F3, 32) (dual of [2197, 2047, 33]-code), using
- construction XX applied to Ce(31) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3148, 2187, F3, 32) (dual of [2187, 2039, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- 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(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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
- discarding factors / shortening the dual code based on linear OA(3150, 2197, F3, 32) (dual of [2197, 2047, 33]-code), using
(150−32, 150, 101030)-Net in Base 3 — Upper bound on s
There is no (118, 150, 101031)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 369999 103710 792856 546977 292607 837678 739457 679042 043493 630622 159069 979873 > 3150 [i]