Best Known (63−13, 63, s)-Nets in Base 3
(63−13, 63, 464)-Net over F3 — Constructive and digital
Digital (50, 63, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (50, 64, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 16, 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, 16, 116)-net over F81, using
(63−13, 63, 1190)-Net over F3 — Digital
Digital (50, 63, 1190)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(363, 1190, F3, 13) (dual of [1190, 1127, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(363, 2195, F3, 13) (dual of [2195, 2132, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([1,6]) [i] based on
- linear OA(357, 2188, F3, 13) (dual of [2188, 2131, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(356, 2188, F3, 6) (dual of [2188, 2132, 7]-code), using the narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(36, 7, F3, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,3)), using
- dual of repetition code with length 7 [i]
- construction X applied to C([0,6]) ⊂ C([1,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(363, 2195, F3, 13) (dual of [2195, 2132, 14]-code), using
(63−13, 63, 127475)-Net in Base 3 — Upper bound on s
There is no (50, 63, 127476)-net in base 3, because
- 1 times m-reduction [i] would yield (50, 62, 127476)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 381531 557144 763863 731075 389049 > 362 [i]