Best Known (157−32, 157, s)-Nets in Base 3
(157−32, 157, 688)-Net over F3 — Constructive and digital
Digital (125, 157, 688)-net over F3, using
- 31 times duplication [i] based on digital (124, 156, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 39, 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, 39, 172)-net over F81, using
(157−32, 157, 1795)-Net over F3 — Digital
Digital (125, 157, 1795)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3157, 1795, F3, 32) (dual of [1795, 1638, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 2219, F3, 32) (dual of [2219, 2062, 33]-code), using
- construction XX applied to Ce(31) ⊂ Ce(27) ⊂ Ce(25) [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(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(36, 29, F3, 3) (dual of [29, 23, 4]-code or 29-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, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(31) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3157, 2219, F3, 32) (dual of [2219, 2062, 33]-code), using
(157−32, 157, 163387)-Net in Base 3 — Upper bound on s
There is no (125, 157, 163388)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 809 174120 446000 075190 379565 017018 672103 676474 751778 014580 958958 543529 932417 > 3157 [i]