Best Known (247−45, 247, s)-Nets in Base 3
(247−45, 247, 1480)-Net over F3 — Constructive and digital
Digital (202, 247, 1480)-net over F3, using
- 1 times m-reduction [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
(247−45, 247, 4488)-Net over F3 — Digital
Digital (202, 247, 4488)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3247, 4488, F3, 45) (dual of [4488, 4241, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3247, 6591, F3, 45) (dual of [6591, 6344, 46]-code), using
- construction X applied to Ce(45) ⊂ Ce(40) [i] based on
- linear OA(3241, 6561, F3, 46) (dual of [6561, 6320, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(45) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3247, 6591, F3, 45) (dual of [6591, 6344, 46]-code), using
(247−45, 247, 979256)-Net in Base 3 — Upper bound on s
There is no (202, 247, 979257)-net in base 3, because
- 1 times m-reduction [i] would yield (202, 246, 979257)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2354 128383 927420 705080 230064 404381 328893 453038 250843 264098 809214 071356 746697 924649 552749 587348 334683 449304 169196 071497 > 3246 [i]