Best Known (101, 101+28, s)-Nets in Base 3
(101, 101+28, 600)-Net over F3 — Constructive and digital
Digital (101, 129, 600)-net over F3, using
- 31 times duplication [i] based on digital (100, 128, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 32, 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, 32, 150)-net over F81, using
(101, 101+28, 1154)-Net over F3 — Digital
Digital (101, 129, 1154)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3129, 1154, F3, 28) (dual of [1154, 1025, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3129, 2197, F3, 28) (dual of [2197, 2068, 29]-code), using
- construction XX applied to Ce(27) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- 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(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(31, 9, F3, 1) (dual of [9, 8, 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
- 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(27) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3129, 2197, F3, 28) (dual of [2197, 2068, 29]-code), using
(101, 101+28, 75280)-Net in Base 3 — Upper bound on s
There is no (101, 129, 75281)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 35 375049 533212 280245 517163 055143 337894 728950 174619 264346 685721 > 3129 [i]