Best Known (206, 206+42, s)-Nets in Base 3
(206, 206+42, 1480)-Net over F3 — Constructive and digital
Digital (206, 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
(206, 206+42, 6640)-Net over F3 — Digital
Digital (206, 248, 6640)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3248, 6640, F3, 42) (dual of [6640, 6392, 43]-code), using
- construction X applied to Ce(42) ⊂ Ce(31) [i] based on
- linear OA(3225, 6561, F3, 43) (dual of [6561, 6336, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(42) ⊂ Ce(31) [i] based on
(206, 206+42, 1870822)-Net in Base 3 — Upper bound on s
There is no (206, 248, 1870823)-net in base 3, because
- 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]