Best Known (73, 73+20, s)-Nets in Base 3
(73, 73+20, 464)-Net over F3 — Constructive and digital
Digital (73, 93, 464)-net over F3, using
- 31 times duplication [i] based on digital (72, 92, 464)-net over F3, using
- t-expansion [i] based on digital (71, 92, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 23, 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, 23, 116)-net over F81, using
- t-expansion [i] based on digital (71, 92, 464)-net over F3, using
(73, 73+20, 1097)-Net over F3 — Digital
Digital (73, 93, 1097)-net over F3, using
- 31 times duplication [i] based on digital (72, 92, 1097)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(392, 1097, F3, 2, 20) (dual of [(1097, 2), 2102, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(392, 2194, F3, 20) (dual of [2194, 2102, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(392, 2187, F3, 20) (dual of [2187, 2095, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(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(19) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(392, 2194, F3, 20) (dual of [2194, 2102, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(392, 1097, F3, 2, 20) (dual of [(1097, 2), 2102, 21]-NRT-code), using
(73, 73+20, 61959)-Net in Base 3 — Upper bound on s
There is no (73, 93, 61960)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 235 676783 101683 549846 314690 830754 083539 802161 > 393 [i]