Best Known (157, 157+29, s)-Nets in Base 3
(157, 157+29, 1480)-Net over F3 — Constructive and digital
Digital (157, 186, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (157, 188, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 47, 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, 47, 370)-net over F81, using
(157, 157+29, 10126)-Net over F3 — Digital
Digital (157, 186, 10126)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3186, 10126, F3, 29) (dual of [10126, 9940, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3186, 19698, F3, 29) (dual of [19698, 19512, 30]-code), using
- (u, u+v)-construction [i] based on
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(3186, 19698, F3, 29) (dual of [19698, 19512, 30]-code), using
(157, 157+29, 6098785)-Net in Base 3 — Upper bound on s
There is no (157, 186, 6098786)-net in base 3, because
- 1 times m-reduction [i] would yield (157, 185, 6098786)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 18511 114686 438793 435064 747389 549159 549379 533521 058708 368946 613459 053963 611762 317092 601037 > 3185 [i]