Best Known (239−49, 239, s)-Nets in Base 3
(239−49, 239, 688)-Net over F3 — Constructive and digital
Digital (190, 239, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (190, 244, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 61, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 61, 172)-net over F81, using
(239−49, 239, 2301)-Net over F3 — Digital
Digital (190, 239, 2301)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3239, 2301, F3, 49) (dual of [2301, 2062, 50]-code), using
- 99 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0) [i] based on linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- 99 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0) [i] based on linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using
(239−49, 239, 264086)-Net in Base 3 — Upper bound on s
There is no (190, 239, 264087)-net in base 3, because
- 1 times m-reduction [i] would yield (190, 238, 264087)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 358814 684994 593704 292981 547881 004659 848533 668248 272464 793435 230734 820751 276078 758126 546055 927236 755110 284961 502033 > 3238 [i]