Best Known (62, 62+19, s)-Nets in Base 3
(62, 62+19, 400)-Net over F3 — Constructive and digital
Digital (62, 81, 400)-net over F3, using
- 31 times duplication [i] based on digital (61, 80, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F81, using
(62, 62+19, 616)-Net over F3 — Digital
Digital (62, 81, 616)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(381, 616, F3, 19) (dual of [616, 535, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(381, 758, F3, 19) (dual of [758, 677, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(373, 730, F3, 19) (dual of [730, 657, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(349, 730, F3, 13) (dual of [730, 681, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(381, 758, F3, 19) (dual of [758, 677, 20]-code), using
(62, 62+19, 36115)-Net in Base 3 — Upper bound on s
There is no (62, 81, 36116)-net in base 3, because
- 1 times m-reduction [i] would yield (62, 80, 36116)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 147 816678 675275 584523 228467 679209 317929 > 380 [i]