Best Known (232−39, 232, s)-Nets in Base 3
(232−39, 232, 1480)-Net over F3 — Constructive and digital
Digital (193, 232, 1480)-net over F3, using
- 4 times m-reduction [i] based on digital (193, 236, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 59, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 59, 370)-net over F81, using
(232−39, 232, 6640)-Net over F3 — Digital
Digital (193, 232, 6640)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3232, 6640, F3, 39) (dual of [6640, 6408, 40]-code), using
- construction X applied to Ce(39) ⊂ Ce(28) [i] based on
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(323, 79, F3, 9) (dual of [79, 56, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(323, 82, F3, 9) (dual of [82, 59, 10]-code), using
- a “GraX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(323, 82, F3, 9) (dual of [82, 59, 10]-code), using
- construction X applied to Ce(39) ⊂ Ce(28) [i] based on
(232−39, 232, 2505949)-Net in Base 3 — Upper bound on s
There is no (193, 232, 2505950)-net in base 3, because
- 1 times m-reduction [i] would yield (193, 231, 2505950)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 164 063450 395714 323069 962368 851687 756450 002226 947917 530114 343736 760511 671386 681102 658045 070090 690118 576406 941921 > 3231 [i]