Best Known (161, 161+31, s)-Nets in Base 3
(161, 161+31, 1480)-Net over F3 — Constructive and digital
Digital (161, 192, 1480)-net over F3, using
- t-expansion [i] based on digital (160, 192, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 48, 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, 48, 370)-net over F81, using
(161, 161+31, 9736)-Net over F3 — Digital
Digital (161, 192, 9736)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3192, 9736, F3, 2, 31) (dual of [(9736, 2), 19280, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3192, 9865, F3, 2, 31) (dual of [(9865, 2), 19538, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3192, 19730, F3, 31) (dual of [19730, 19538, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3192, 19731, F3, 31) (dual of [19731, 19539, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- 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(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(311, 47, F3, 5) (dual of [47, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3192, 19731, F3, 31) (dual of [19731, 19539, 32]-code), using
- OOA 2-folding [i] based on linear OA(3192, 19730, F3, 31) (dual of [19730, 19538, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(3192, 9865, F3, 2, 31) (dual of [(9865, 2), 19538, 32]-NRT-code), using
(161, 161+31, 3820137)-Net in Base 3 — Upper bound on s
There is no (161, 192, 3820138)-net in base 3, because
- 1 times m-reduction [i] would yield (161, 191, 3820138)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13 494625 212779 376688 172065 374147 055084 629217 291294 375668 051162 702484 560544 256586 726949 123961 > 3191 [i]