Best Known (83, 83+23, s)-Nets in Base 3
(83, 83+23, 464)-Net over F3 — Constructive and digital
Digital (83, 106, 464)-net over F3, using
- 2 times m-reduction [i] based on digital (83, 108, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 27, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 27, 116)-net over F81, using
(83, 83+23, 1097)-Net over F3 — Digital
Digital (83, 106, 1097)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3106, 1097, F3, 2, 23) (dual of [(1097, 2), 2088, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3106, 2194, F3, 23) (dual of [2194, 2088, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 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(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(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3106, 2194, F3, 23) (dual of [2194, 2088, 24]-code), using
(83, 83+23, 87957)-Net in Base 3 — Upper bound on s
There is no (83, 106, 87958)-net in base 3, because
- 1 times m-reduction [i] would yield (83, 105, 87958)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 125 248709 185138 159386 344036 446653 322192 675407 574737 > 3105 [i]