Best Known (169, 169+33, s)-Nets in Base 3
(169, 169+33, 1480)-Net over F3 — Constructive and digital
Digital (169, 202, 1480)-net over F3, using
- 2 times m-reduction [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
(169, 169+33, 9103)-Net over F3 — Digital
Digital (169, 202, 9103)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3202, 9103, F3, 2, 33) (dual of [(9103, 2), 18004, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3202, 9852, F3, 2, 33) (dual of [(9852, 2), 19502, 34]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3201, 9852, F3, 2, 33) (dual of [(9852, 2), 19503, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3201, 19704, F3, 33) (dual of [19704, 19503, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3200, 19703, F3, 33) (dual of [19703, 19503, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(3199, 19684, F3, 33) (dual of [19684, 19485, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3181, 19684, F3, 31) (dual of [19684, 19503, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(31, 19, F3, 1) (dual of [19, 18, 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
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3200, 19703, F3, 33) (dual of [19703, 19503, 34]-code), using
- OOA 2-folding [i] based on linear OA(3201, 19704, F3, 33) (dual of [19704, 19503, 34]-code), using
- 31 times duplication [i] based on linear OOA(3201, 9852, F3, 2, 33) (dual of [(9852, 2), 19503, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3202, 9852, F3, 2, 33) (dual of [(9852, 2), 19502, 34]-NRT-code), using
(169, 169+33, 3352296)-Net in Base 3 — Upper bound on s
There is no (169, 202, 3352297)-net in base 3, because
- 1 times m-reduction [i] would yield (169, 201, 3352297)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 796842 115670 322448 054544 714814 830932 298807 850278 821439 987180 470681 982281 926367 899296 916475 590465 > 3201 [i]