Best Known (89, 89+24, s)-Nets in Base 3
(89, 89+24, 600)-Net over F3 — Constructive and digital
Digital (89, 113, 600)-net over F3, using
- 31 times duplication [i] based on digital (88, 112, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 28, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- trace code for nets [i] based on digital (4, 28, 150)-net over F81, using
(89, 89+24, 1196)-Net over F3 — Digital
Digital (89, 113, 1196)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3113, 1196, F3, 24) (dual of [1196, 1083, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3113, 2194, F3, 24) (dual of [2194, 2081, 25]-code), using
- 1 times truncation [i] based on linear OA(3114, 2195, F3, 25) (dual of [2195, 2081, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 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(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(3114, 2195, F3, 25) (dual of [2195, 2081, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3113, 2194, F3, 24) (dual of [2194, 2081, 25]-code), using
(89, 89+24, 82255)-Net in Base 3 — Upper bound on s
There is no (89, 113, 82256)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 821793 193084 174400 096384 575140 912110 503731 451638 058369 > 3113 [i]