Best Known (206, 206+43, s)-Nets in Base 3
(206, 206+43, 1480)-Net over F3 — Constructive and digital
Digital (206, 249, 1480)-net over F3, using
- 31 times duplication [i] based on digital (205, 248, 1480)-net over F3, using
- t-expansion [i] based on digital (202, 248, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 62, 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, 62, 370)-net over F81, using
- t-expansion [i] based on digital (202, 248, 1480)-net over F3, using
(206, 206+43, 6168)-Net over F3 — Digital
Digital (206, 249, 6168)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3249, 6168, F3, 43) (dual of [6168, 5919, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 6634, F3, 43) (dual of [6634, 6385, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,16]) [i] based on
- linear OA(3225, 6562, F3, 43) (dual of [6562, 6337, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(324, 72, F3, 9) (dual of [72, 48, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(324, 80, F3, 9) (dual of [80, 56, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(324, 80, F3, 9) (dual of [80, 56, 10]-code), using
- construction X applied to C([0,21]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 6634, F3, 43) (dual of [6634, 6385, 44]-code), using
(206, 206+43, 1870822)-Net in Base 3 — Upper bound on s
There is no (206, 249, 1870823)-net in base 3, because
- 1 times m-reduction [i] would yield (206, 248, 1870823)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 21187 154993 936757 633695 969274 359415 966346 198286 288707 909264 143501 535575 739594 527417 670401 332127 060660 834180 836237 615847 > 3248 [i]