Best Known (221−36, 221, s)-Nets in Base 3
(221−36, 221, 1480)-Net over F3 — Constructive and digital
Digital (185, 221, 1480)-net over F3, using
- t-expansion [i] based on digital (184, 221, 1480)-net over F3, using
- 3 times m-reduction [i] based on digital (184, 224, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 56, 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, 56, 370)-net over F81, using
- 3 times m-reduction [i] based on digital (184, 224, 1480)-net over F3, using
(221−36, 221, 9621)-Net over F3 — Digital
Digital (185, 221, 9621)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3221, 9621, F3, 2, 36) (dual of [(9621, 2), 19021, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3221, 9853, F3, 2, 36) (dual of [(9853, 2), 19485, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3221, 19706, F3, 36) (dual of [19706, 19485, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(3217, 19684, F3, 37) (dual of [19684, 19467, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 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(34, 22, F3, 2) (dual of [22, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- OOA 2-folding [i] based on linear OA(3221, 19706, F3, 36) (dual of [19706, 19485, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(3221, 9853, F3, 2, 36) (dual of [(9853, 2), 19485, 37]-NRT-code), using
(221−36, 221, 2723240)-Net in Base 3 — Upper bound on s
There is no (185, 221, 2723241)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2778 424007 421171 649546 101207 735934 099440 622431 899776 222135 390197 111151 734751 383117 544542 636144 046259 753305 > 3221 [i]