Best Known (128−26, 128, s)-Nets in Base 3
(128−26, 128, 640)-Net over F3 — Constructive and digital
Digital (102, 128, 640)-net over F3, using
- t-expansion [i] based on digital (101, 128, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 32, 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, 32, 160)-net over F81, using
(128−26, 128, 1619)-Net over F3 — Digital
Digital (102, 128, 1619)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3128, 1619, F3, 26) (dual of [1619, 1491, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3128, 2217, F3, 26) (dual of [2217, 2089, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- 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(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(392, 2187, F3, 20) (dual of [2187, 2095, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), 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(25) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3128, 2217, F3, 26) (dual of [2217, 2089, 27]-code), using
(128−26, 128, 141301)-Net in Base 3 — Upper bound on s
There is no (102, 128, 141302)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 11 790631 049212 617004 846104 073870 660777 084026 939721 564594 663989 > 3128 [i]