Best Known (207−33, 207, s)-Nets in Base 3
(207−33, 207, 1480)-Net over F3 — Constructive and digital
Digital (174, 207, 1480)-net over F3, using
- t-expansion [i] based on digital (172, 207, 1480)-net over F3, using
- 1 times m-reduction [i] based on digital (172, 208, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 52, 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, 52, 370)-net over F81, using
- 1 times m-reduction [i] based on digital (172, 208, 1480)-net over F3, using
(207−33, 207, 9862)-Net over F3 — Digital
Digital (174, 207, 9862)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3207, 9862, F3, 2, 33) (dual of [(9862, 2), 19517, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3207, 19724, F3, 33) (dual of [19724, 19517, 34]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(3207, 19724, F3, 33) (dual of [19724, 19517, 34]-code), using
(207−33, 207, 4725453)-Net in Base 3 — Upper bound on s
There is no (174, 207, 4725454)-net in base 3, because
- 1 times m-reduction [i] would yield (174, 206, 4725454)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 193 633175 308265 066813 744181 627537 704061 503664 368065 397468 530864 640308 064817 027516 336758 099853 406113 > 3206 [i]