Best Known (83, 105, s)-Nets in Base 3
(83, 105, 600)-Net over F3 — Constructive and digital
Digital (83, 105, 600)-net over F3, using
- 31 times duplication [i] based on digital (82, 104, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 26, 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, 26, 150)-net over F81, using
(83, 105, 1239)-Net over F3 — Digital
Digital (83, 105, 1239)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3105, 1239, F3, 22) (dual of [1239, 1134, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3105, 2208, F3, 22) (dual of [2208, 2103, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- 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(385, 2187, F3, 19) (dual of [2187, 2102, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(378, 2187, F3, 17) (dual of [2187, 2109, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(34, 19, F3, 2) (dual of [19, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), 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(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3105, 2208, F3, 22) (dual of [2208, 2103, 23]-code), using
(83, 105, 87957)-Net in Base 3 — Upper bound on s
There is no (83, 105, 87958)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 125 248709 185138 159386 344036 446653 322192 675407 574737 > 3105 [i]