Best Known (204−33, 204, s)-Nets in Base 3
(204−33, 204, 1480)-Net over F3 — Constructive and digital
Digital (171, 204, 1480)-net over F3, using
- t-expansion [i] based on digital (169, 204, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 51, 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, 51, 370)-net over F81, using
(204−33, 204, 9797)-Net over F3 — Digital
Digital (171, 204, 9797)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3204, 9797, F3, 2, 33) (dual of [(9797, 2), 19390, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3204, 9854, F3, 2, 33) (dual of [(9854, 2), 19504, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3204, 19708, F3, 33) (dual of [19708, 19504, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3204, 19709, F3, 33) (dual of [19709, 19505, 34]-code), using
- construction XX applied to Ce(33) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(31, 22, F3, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(31, 4, F3, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to Ce(33) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3204, 19709, F3, 33) (dual of [19709, 19505, 34]-code), using
- OOA 2-folding [i] based on linear OA(3204, 19708, F3, 33) (dual of [19708, 19504, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(3204, 9854, F3, 2, 33) (dual of [(9854, 2), 19504, 34]-NRT-code), using
(204−33, 204, 3845766)-Net in Base 3 — Upper bound on s
There is no (171, 204, 3845767)-net in base 3, because
- 1 times m-reduction [i] would yield (171, 203, 3845767)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 7 171596 370570 258645 513015 084162 092070 465189 014108 658873 244632 050006 773159 771460 868632 350402 488545 > 3203 [i]